7.4: Singular Value Decompositions
The Spectral Theorem has animated the past few sections. In particular, we applied the fact that symmetric matrices can be orthogonally diagonalized to simplify quadratic forms, which enabled us to use principal component analysis to reduce the dimension of a dataset.
But what can we do with matrices that are not symmetric or even square? For instance, the following matrices are not diagonalizable, much less orthogonally so:
In this section, we will develop a description of matrices called the singular value decomposition that is, in many ways, analogous to an orthogonal diagonalization. For example, we have seen that any symmetric matrix can be written in the form \(QDQ^T\) where \(Q\) is an orthogonal matrix and \(D\) is diagonal. A singular value decomposition will have the form \(U\Sigma V^T\) where \(U\) and \(V\) are orthogonal and \(\Sigma\) is diagonal. Most notably, we will see that every matrix has a singular value decomposition whether it's symmetric or not.
Preview Activity 7.4.1.
Let's review orthogonal diagonalizations and quadratic forms as our understanding of singular value decompositions will rely on them.
- Suppose that \(A\) is any matrix. Explain why the matrix \(G = A^TA\) is symmetric.
- Suppose that \(A = \begin{bmatrix} 1 & 2 \\ -2 & -1 \\ \end{bmatrix}\text{.}\) Find the matrix \(G=A^TA\) and write out the quadratic form \(q_G\left(\twovec{x_1}{x_2}\right)\) as a function of \(x_1\) and \(x_2\text{.}\)
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What is the maximum value of \(q_G(\mathbf x)\) and in which direction does it occur?
- What is the minimum value of \(q_G(\mathbf x)\) and in which direction does it occur?
- What is the geometric relationship between the directions in which the maximum and minimum values occur?
Finding singular value decompositions
We will begin by explaining what a singular value decomposition is and how we can find one for a given matrix \(A\text{.}\)
Recall how the orthogonal diagonalization of a symmetric matrix is formed: if \(A\) is symmetric, we write \(A = QDQ^T\) where the diagonal entries of \(D\) are the eigenvalues of \(A\) and the columns of \(Q\) are the associated eigenvectors. Moreover, the eigenvalues are related to the maximum and minimum values of the associated quadratic form \(q_A(\mathbf u)\) among all unit vectors.
A general matrix, particularly a matrix that is not square, may not have eigenvalues and eigenvectors, but we can discover analogous features, called singular values and singular vectors , by studying a function somewhat similar to a quadratic form. More specifically, any matrix \(A\) defines a function
which measures the length of \(A\mathbf x\text{.}\) For example, the diagonal matrix \(D=\begin{bmatrix} 3 & 0 \\ 0 & -2 \\ \end{bmatrix}\) gives the function \(l_D(\mathbf x) = \sqrt{9x_1^2 + 4x_2^2}\text{.}\) The presence of the square root means that this function is not a quadratic form. We can, however, define the singular values and vectors by looking for the maximum and minimum of this function \(l_A(\mathbf u)\) among all unit vectors \(\mathbf u\text{.}\)
While \(l_A(\mathbf x)\) is not itself a quadratic form, it becomes one if we square it:
We call \(G=A^TA\text{,}\) the Gram matrix associated to \(A\) and note that
This is important in the next activity, which introduces singular values and singular vectors.
Activity 7.4.2.
The following interactive figure will help us explore singular values and vectors geometrically before we begin a more algebraic approach. This figure is also available at gvsu.edu/s/0YE .
Select the matrix \(A=\begin{bmatrix} 1 & 2 \\ -2 & - 1 \\ \end{bmatrix}\text{.}\) As we vary the vector \(\mathbf x\text{,}\) we see the vector \(A\mathbf x\) in gray while the height of the blue bar to the right tells us \(l_A(\mathbf x) = |A\mathbf x|\text{.}\)
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The first
singular value
\(\sigma_1\) is the maximum value of \(l_A(\mathbf x)\) and an associated
right singular vector
\(\mathbf v_1\) is a unit vector describing a direction in which this maximum occurs.
Use the diagram to find the first singular value \(\sigma_1\) and an associated right singular vector \(\mathbf v_1\text{.}\)
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The second singular value \(\sigma_2\) is the minimum value of \(l_A(\mathbf x)\) and an associated right singular vector \(\mathbf v_2\) is a unit vector describing a direction in which this minimum occurs.
Use the diagram to find the second singular value \(\sigma_2\) and an associated right singular vector \(\mathbf v_2\text{.}\)
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Here's how we can find the right singular values and vectors without using the diagram. Remember that \(l_A(\mathbf x) = \sqrt{q_G(\mathbf x)}\) where \(G=A^TA\) is the Gram matrix associated to \(A\text{.}\) Since \(G\) is symmetric, it is orthogonally diagonalizable. Find \(G\) and an orthogonal diagonalization of it.
What is the maximum value of the quadratic form \(q_G(\mathbf x)\) among all unit vectors and in which direction does it occur? What is the minimum value of \(q_G(\mathbf x)\) and in which direction does it occur?
- Because \(l_A(\mathbf x) = \sqrt{q_G(\mathbf x)}\text{,}\) the first singular value \(\sigma_1\) will be the square root of the maximum value of \(q_G(\mathbf x)\) and \(\sigma_2\) the square root of the minimum. Verify that the singular values that you found from the diagram are the square roots of the maximum and minimum values of \(q_G(\mathbf x)\text{.}\)
- Verify that the right singular vectors \(\mathbf v_1\) and \(\mathbf v_2\) that you found from the diagram are the directions in which the maximum and minimum values occur.
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Finally, we introduce the
left singular vectors
\(\mathbf u_1\) and \(\mathbf u_2\) by requiring that \(A\mathbf v_1 = \sigma_1\mathbf u_1\) and \(A\mathbf v_2=\sigma_2\mathbf u_2\text{.}\) Find the two left singular vectors.
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Form the matrices
\begin{equation*} U = \begin{bmatrix}\mathbf u_1 & \mathbf u_2 \end{bmatrix}, \hspace{24pt} \Sigma = \begin{bmatrix} \sigma_1 & 0 \\ 0 & \sigma_2 \\ \end{bmatrix}, \hspace{24pt} V = \begin{bmatrix}\mathbf v_1 & \mathbf v_2 \end{bmatrix} \end{equation*}
and explain why \(AV = U\Sigma\text{.}\)
- Finally, explain why \(A=U\Sigma V^T\) and verify that this relationship holds for this specific example.
As this activity shows, the singular values of \(A\) are the maximum and minimum values of \(l_A(\mathbf x)=|A\mathbf x|\) among all unit vectors and the right singular vectors \(\mathbf v_1\) and \(\mathbf v_2\) are the directions in which they occur. The key to finding the singular values and vectors is to utilize the Gram matrix \(G\) and its associated quadratic form \(q_G(\mathbf x)\text{.}\) We will illustrate with some more examples.
We will find a singular value decomposition of the matrix \(A=\begin{bmatrix} 1 & 2 \\ -1 & 2 \end{bmatrix} \text{.}\) Notice that this matrix is not symmetric so it cannot be orthogonally diagonalized.
We begin by constructing the Gram matrix \(G = A^TA = \begin{bmatrix} 2 & 0 \\ 0 & 8 \\ \end{bmatrix}\text{.}\) Since \(G\) is symmetric, it can be orthogonally diagonalized with
We now know that the maximum value of the quadratic form \(q_G(\mathbf x)\) is 8, which occurs in the direction \(\twovec01\text{.}\) Since \(l_A(\mathbf x) = \sqrt{q_G(\mathbf x)}\text{,}\) this tells us that the maximum value of \(l_A(\mathbf x)\text{,}\) the first singular value, is \(\sigma_1=\sqrt{8}\) and that this occurs in the direction of the first right singular vector \(\mathbf v_1=\twovec01\text{.}\)
In the same way, we also know that the second singular value \(\sigma_2=\sqrt{2}\) with associated right singular vector \(\mathbf v_2=\twovec10\text{.}\)
The first left singular vector \(\mathbf u_1\) is defined by \(A\mathbf v_1 = \twovec22 = \sigma_1\mathbf u_1\text{.}\) Because \(\sigma_1 = \sqrt{8}\text{,}\) we have \(\mathbf u_1 = \twovec{1/\sqrt{2}}{1/\sqrt{2}}\text{.}\) Notice that \(\mathbf u_1\) is a unit vector because \(\sigma_1 = |A\mathbf v_1|\text{.}\)
In the same way, the second left singular vector is defined by \(A\mathbf v_2 = \twovec1{-1} = \sigma_2\mathbf u_2\text{,}\) which gives us \(\mathbf u_2 = \twovec{1/\sqrt{2}}{-1/\sqrt{2}}\text{.}\)
We then construct
We now have \(AV=U\Sigma\) because
Because the right singular vectors, the columns of \(V\text{,}\) are eigenvectors of the symmetric matrix \(G\text{,}\) they form an orthonormal basis, which means that \(V\) is orthogonal. Therefore, we have \((AV)V^T = A = U\Sigma V^T\text{.}\) This gives the singular value decomposition
To summarize, we find a singular value decomposition of a matrix \(A\) in the following way:
- Construct the Gram matrix \(G=A^TA\) and find an orthogonal diagonalization to obtain eigenvalues \(\lambda_i\) and an orthonormal basis of eigenvectors.
- The singular values of \(A\) are the squares roots of eigenvalues \(\lambda_i\) of \(G\text{;}\) that is, \(\sigma_i = \sqrt{\lambda_i}\text{.}\) For reasons we'll see in the next section, the singular values are listed in decreasing order: \(\sigma_1 \geq \sigma_2 \geq \ldots \text{.}\) The right singular vectors \(\mathbf v_i\) are the associated eigenvectors of \(G\text{.}\)
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The left singular vectors \(\mathbf u_i\) are found by \(A\mathbf v_i = \sigma_i\mathbf u_i\text{.}\) Because \(\sigma_i=|A\mathbf v_i|\text{,}\) we know that \(\mathbf u_i\) will be a unit vector.
In fact, the left singular vectors will also form an orthonormal basis. To see this, suppose that the associcated singular values are nonzero. We then have:
\begin{align*} \sigma_i\sigma_j(\mathbf u_i\cdot\mathbf u_j) & {}={} (\sigma_i\mathbf u_i)\cdot(\sigma_j\mathbf u_j) = (A\mathbf v_i)\cdot(A\mathbf v_j)\\ & {}={} \mathbf v_i\cdot(A^TA\mathbf v_j) \\ & {}={} \mathbf v_i\cdot(G\mathbf v_j) = \lambda_j\mathbf v_i\cdot\mathbf v_j = 0 \end{align*}since the right singular vectors are orthogonal.
Let's find a singular value decomposition for the symmetric matrix \(A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\text{.}\) The associated Gram matrix is
which has an orthogonal diagonalization with
This gives singular values and vectors
and the singular value decomposition \(A=U\Sigma V^T\) where
This example is special because \(A\) is symmetric. With a little thought, it's possible to relate this singular value decomposition to an orthogonal diagonalization of \(A\) using the fact that \(G=A^TA = A^2\text{.}\)
Activity 7.4.3.
In this activity, we will construct the singular value decomposition of \(A=\begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}\text{.}\) Notice that this matrix is not square so there are no eigenvalues and eigenvectors associated to it.
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Construct the Gram matrix \(G=A^TA\) and find an orthogonal diagonalization of it.
- Identify the singular values of \(A\) and the right singular vectors \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) and \(\mathbf v_3\text{.}\) What is the dimension of these vectors? How many nonzero singular values are there?
- Find the left singular vectors \(\mathbf u_1\) and \(\mathbf u_2\) using the fact that \(A\mathbf v_i = \sigma_i\mathbf u_i\text{.}\) What is the dimension of these vectors? What happens if you try to find a third left singular vector \(\mathbf u_3\) in this way?
- As before, form the orthogonal matrices \(U\) and \(V\) from the left and right singular vectors. What are the dimensions of \(U\) and \(V\text{?}\) How do these dimensions relate to the number of rows and columns of \(A\text{?}\)
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Now form \(\Sigma\) so that it has the same shape as \(A\text{:}\)
\begin{equation*} \Sigma = \begin{bmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \end{bmatrix} \end{equation*}
and verify that \(A = U\Sigma V^T\text{.}\)
- How can you use this singular value decomposition of \(A=U\Sigma V^T\) to easily find a singular value decomposition of \(A^T=\begin{bmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \\ \end{bmatrix}\text{?}\)
We will find a singular value decomposition of the matrix \(A=\begin{bmatrix} 2 & -2 & 1 \\ -4 & -8 & -8 \\ \end{bmatrix}\text{.}\)
Finding an orthogonal diagonalization of \(G=A^TA\) gives
which gives singular values \(\sigma_1=\sqrt{144}=12\text{,}\) \(\sigma_2 = \sqrt{9}= 3\text{,}\) and \(\sigma_3 = 0\text{.}\) The right singular vectors \(\mathbf v_i\) appear as the columns of \(Q\) so that \(V = Q\text{.}\)
We now find
Notice that it's not possible to find a third left singular vector since \(A\mathbf v_3=\zerovec\text{.}\) We thereform form the matrices
which gives the singular value decomposition \(A=U\Sigma V^T\text{.}\)
Notice that \(U\) is a \(2\times2\) orthogonal matrix because \(A\) has two rows, and \(V\) is a \(3\times3\) orthogonal matrix because \(A\) has three columns.
As we'll see in the next section, some additional work may be needed to construct the left singular vectors \(\mathbf u_j\) if more of the singular values are zero, but we won't worry about that now. For the time being, let's record our work in the following theorem.
Theorem 7.4.5. The singular value decomposition.
An \(m\times n\) matrix \(A\) may be written as \(A=U\Sigma V^T\) where \(U\) is an orthogonal \(m\times m\) matrix, \(V\) is an orthogonal \(n\times n\) matrix, and \(\Sigma\) is an \(m\times n\) matrix whose entries are zero except for the singular values of \(A\) which appear in decreasing order on the diagonal.Notice that a singular value decomposition of \(A\) gives us a singular value decomposition of \(A^T\text{.}\) More specifically, if \(A=U\Sigma V^T\text{,}\) then
If \(A=U\Sigma V^T\text{,}\) then \(A^T = V\Sigma^T U^T\text{.}\) In other words, \(A\) and \(A^T\) share the same singular values, and the left singular vectors of \(A\) are the right singular vectors of \(A^T\) and vice-versa.
As we said earlier, the singular value decomposition should be thought of a generalization of an orthogonal diagonalization. For instance, the Spectral Theorem tells us that a symmetric matrix can be written as \(QDQ^T\text{.}\) Many matrices, however, are not symmetric and so they are not orthogonally diagonalizable. However, every matrix has a singular value decomposition \(U\Sigma V^T\text{.}\) The price of this generalization is that we usually have two sets of singular vectors that form the orthogonal matrices \(U\) and \(V\) whereas a symmetric matrix has a single set of eignevectors that form the orthogonal matrix \(Q\text{.}\)
The structure of singular value decompositions
Now that we have an understanding of what a singular value decomposition is and how to construct it, let's explore the ways in which a singular value decomposition reveals the underlying structure of the matrix. As we'll see, the matrices \(U\) and \(V\) in a singular value decomposition provide convenient bases for some important subspaces, such as the column and null spaces of the matrix. This observation will provide the key to some of our uses of these decompositions in the next section.
Activity 7.4.4.
Let's suppose that a matrix \(A\) has a singular value decomposition \(A=U\Sigma V^T\) where
- What are the dimensions of \(A\text{;}\) that is, how many rows and columns does \(A\) have?
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Suppose we write a three-dimensional vector \(\mathbf x\) as a linear combination of right singular vectors:
\begin{equation*} \mathbf x = c_1\mathbf v_1 + c_2\mathbf v_2 + c_3\mathbf v_3\text{.} \end{equation*}
We would like to find an expression for \(A\mathbf x\text{.}\)
To begin, \(V^T\mathbf x = \threevec{\mathbf v_1\cdot\mathbf x} {\mathbf v_2\cdot\mathbf x} {\mathbf v_3\cdot\mathbf x} = \threevec{c_1}{c_2}{c_3} \text{.}\)
Now \(\Sigma V^T \mathbf x = \begin{bmatrix} 20 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\threevec{c_1}{c_2}{c_3} = \fourvec{20c_1}{5c_2}00\text{.}\)
And finally, \(A\mathbf x = U\Sigma V^T\mathbf x = \begin{bmatrix} \mathbf u_1 & \mathbf u_2 & \mathbf u_3 & \mathbf u_4 \end{bmatrix} \fourvec{20c_1}{5c_2}00 = 20c_1\mathbf u_1 + 5c_2\mathbf u_2\text{.}\)
To summarize, we have \(A\mathbf x = 20c_1\mathbf u_1 + 5c_2\mathbf u_2\text{.}\)
What condition on \(c_1\text{,}\) \(c_2\text{,}\) and \(c_3\) must be satisfied if \(\mathbf x\) is a solution to the equation \(A\mathbf x=40\mathbf u_1 + 20\mathbf u_2\text{?}\) Is there a unique solution or infinitely many?
- Remembering that \(\mathbf u_1\) and \(\mathbf u_2\) are linearly independent, what condition on \(c_1\text{,}\) \(c_2\text{,}\) and \(c_3\) must be satisfied if \(A\mathbf x = \zerovec\text{?}\)
- How do the right singular vectors \(\mathbf v_i\) provide a basis for \(Nul(A)\text{,}\) the subspace of solutions to the equation \(A\mathbf x = \zerovec\text{?}\)
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Remember that \(\mathbf b\) is in \(Col(A)\) if the equation \(A\mathbf x = \mathbf b\) is consistent, which means that
\begin{equation*} A\mathbf x = 20c_1\mathbf u_1 + 5c_2\mathbf u_2 = \mathbf b \end{equation*}
for some coefficients \(c_1\) and \(c_2\text{.}\) How do the left singular vectors \(\mathbf u_i\) provide an orthonormal basis for \(Col(A)\text{?}\)
- Remember that \(Rank(A)\) is the dimension of the column space. What is \(Rank(A)\) and how do the number of nonzero singular values determine \(Rank(A)\text{?}\)
This activity shows how a singular value decomposition of a matrix encodes important information about its null and column spaces. This is, in fact, the key observation that makes singular value decompositions so useful: the left and right singular vectors provide orthonormal bases for \(Nul(A)\) and \(Col(A)\text{.}\)
Suppose we have a singular value decomposition \(A=U\Sigma V^T\) where \(\Sigma = \begin{bmatrix} \sigma_1 & 0 & 0 & 0 & 0 \\ 0 & \sigma_2 & 0 & 0 & 0 \\ 0 & 0 & \sigma_3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \text{.}\) This means that \(A\) has four rows and five columns just as \(\Sigma\) does.
As in the activity, if \(\mathbf x = c_1 \mathbf v_1 + c_2\mathbf v_2 + \ldots + c_5\mathbf v_5\text{,}\) we have
If \(\mathbf b\) is in the \(Col(A)\text{,}\) then \(\mathbf b\) must have the form
which says that \(\mathbf b\) is a linear combination of \(\mathbf u_1\text{,}\) \(\mathbf u_2\text{,}\) and \(\mathbf u_3\text{.}\) These three vectors therefore form a basis for \(Col(A)\text{.}\) In fact, since they are columns in the orthogonal matrix \(U\text{,}\) they form an orthonormal basis for \(Col(A)\text{.}\)
Remembering that \(Rank(A)=dimCol(A)\text{,}\) we see that \(Rank(A) = 3\text{,}\) which results from the three nonzero singular values. In general, the rank \(r\) of a matrix \(A\) equals the number of nonzero singular values, and \(\mathbf u_1, \mathbf u_2, \ldots,\mathbf u_r\) form an orthonormal basis for \(Col(A)\text{.}\)
Moreover, if \(\mathbf x = c_1 \mathbf v_1 + c_2\mathbf v_2 + \ldots + c_5\mathbf v_5\) satisfies \(A\mathbf x = \zerovec\text{,}\) then
which implies that \(c_1=0\text{,}\) \(c_2=0\text{,}\) and \(c_3=0\text{.}\) Therefore, \(\mathbf x = c_4\mathbf v_4+c_5\mathbf v_5\) so \(\mathbf v_4\) and \(\mathbf v_5\) form an orthonormal basis for \(Nul(A)\text{.}\)
More generally, if \(A\) is an \(m\times n\) matrix and if \(Rank(A) = r\text{,}\) the last \(n-r\) right singular vectors form an orthonormal basis for \(Nul(A)\text{.}\)
Generally speaking, if the rank of an \(m\times n\) matrix \(A\) is \(r\text{,}\) then there are \(r\) nonzero singular values and \(\Sigma\) has the form
The first \(r\) columns of \(U\) form an orthonormal basis for \(Col(A)\text{:}\)
and the last \(n-r\) columns of \(V\) form an orthonormal basis for \(Nul(A)\text{:}\)
In fact, we can say more. Remember that Proposition 7.4.6 says that \(A\) and its transpose \(A^T\) share the same singular values. Since the rank of a matrix equals its number of nonzero singular values, this means that \(Rank(A)=Rank(A^T)\text{,}\) a fact that we cited back in Section 6.2 .
For any matrix \(A\text{,}\)
If we have a singular value decomposition of an \(m\times n\) matrix \(A=U\Sigma V^T\text{,}\) Proposition 7.4.6 also tells us that the left singular vectors of \(A\) are the right singular vectors of \(A^T\text{.}\) Therefore, \(U\) is the \(m\times m\) matrix whose columns are the right singular vectors of \(A^T\text{.}\) This means that the last \(m-r\) vectors form an orthonormal basis for \(Nul(A^T)\text{.}\) Therefore, the columns of \(U\) provide orthonormal bases for \(Col(A)\) and \(Nul(A^T)\text{:}\)
This reflects the familiar fact that \(Nul(A^T)\) is the orthogonal complement of \(Col(A)\text{.}\)
In the same way, \(V\) is the \(n\times n\) matrix whose columns are the left singular vectors of \(A^T\text{,}\) which means that the first \(r\) vectors form an orthonormal basis for \(Col(A^T)\text{.}\) Because the columns of \(A^T\) are the rows of \(A\text{,}\) this subspace is sometimes called the row space of \(A\) and denoted \(Row(A)\text{.}\) While we have yet to have an occasion to use \(Row(A)\text{,}\) there are times when it is important to have an orthonormal basis for it. Fortunately, a singular value decomposition provides just that. To summarize, the columns of \(V\) provide orthonormal bases for \(Col(A^T)\) and \(Nul(A)\text{:}\)
Considered altogether, the subspaces \(Col(A)\text{,}\) \(Nul(A)\text{,}\) \(Col(A^T)\text{,}\) and \(Nul(A^T)\) are called the four fundamental subspaces associated to \(A\text{.}\) In addition to telling us the rank of a matrix, a singular value decomposition gives us orthonormal bases for all four fundamental subspaces.
Theorem 7.4.9.
Suppose \(A\) is an \(m\times n\) matrix having a singular value decomposition \(A=U\Sigma V^T\text{.}\) Then
- \(r=Rank(A)\) is the number of nonzero singular values.
- The columns \(\mathbf u_1,\mathbf u_2,\ldots,\mathbf u_r\) form an orthonormal basis for \(Col(A)\text{.}\)
- The columns \(\mathbf u_{r+1},\ldots,\mathbf u_m\) form an orthonormal basis for \(Nul(A^T)\text{.}\)
- The columns \(\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_r\) form an orthonormal basis for \(Col(A^T)\text{.}\)
- The columns \(\mathbf v_{r+1},\ldots,\mathbf v_n\) form an orthonormal basis for \(Nul(A)\text{.}\)
When we previously outlined a procedure for finding a singular decomposition of an \(m\times n\) matrix \(A\text{,}\) we found the left singular vectors \(\mathbf u_j\) using the expression \(A\mathbf v_j = \sigma_j\mathbf u_j\text{.}\) This produces left singular vectors \(\mathbf u_1, \mathbf u_2,\ldots,\mathbf u_r\text{,}\) where \(r=Rank(A)\text{.}\) If \(r\lt m\text{,}\) however, we still need to find the left singular vectors \(\mathbf u_{r+1},\ldots,\mathbf u_m\text{.}\) Theorem 7.4.9 tells us how to do that: because those vectors form an orthonormal basis for \(Nul(A^T)\text{,}\) we can find them by solving \(A^T\mathbf x = \zerovec\) to find a basis for \(Nul(A^T)\) and applying the Gram-Schmidt algorithm.
We won't worry about this issue too much, however, as we will frequently use software to find singular value decompositions for us.
Reduced singular value decompositions
As we'll see in the next section, there are times when it is helpful to express a singular value decomposition in a slightly different form.
Activity 7.4.5.
Suppose we have a singular value decomposition \(A = U\Sigma V^T\) where
- What are the dimensions of \(A\text{?}\) What is \(Rank(A)\text{?}\)
- Identify bases for \(Col(A)\) and \(Col(A^T)\text{.}\)
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Explain why
\begin{equation*} U\Sigma = \begin{bmatrix} \mathbf u_1 & \mathbf u_2 \end{bmatrix} \begin{bmatrix} 18 & 0 & 0 \\ 0 & 4 & 0 \\ \end{bmatrix}\text{.} \end{equation*}
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Explain why
\begin{equation*} \begin{bmatrix} 18 & 0 & 0 \\ 0 & 4 & 0 \\ \end{bmatrix}V^T = \begin{bmatrix} 18 & 0 \\ 0 & 4 \\ \end{bmatrix} \begin{bmatrix} \mathbf v_1 & \mathbf v_2 \end{bmatrix}^T\text{.} \end{equation*}
- If \(A = U\Sigma V^T\text{,}\) explain why \(A=U_r\Sigma_rV_r^T\) where the columns of \(U_r\) are an orthonormal basis for \(Col(A)\text{,}\) \(\Sigma_r\) is a diagonal, invertible matrix, and the columns of \(V_r\) form an orthonormal basis for \(Col(A^T)\text{.}\)
We call this a reduced singular value decomposition .
If \(A\) is an \(m\times n\) matrix having rank \(r\text{,}\) then \(A=U_r \Sigma_r V_r^T\) where
- \(U_r\) is an \(m\times r\) matrix whose columns form an orthonormal basis for \(Col(A)\text{,}\)
- \(\Sigma_r=\begin{bmatrix} \sigma_1 & 0 & \ldots & 0 \\ 0 & \sigma_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \sigma_r \\ \end{bmatrix}\) is an \(r\times r\) diagonal, invertible matrix, and
- \(V_r\) is an \(n\times r\) matrix whose columns form an orthonormal basis for \(Col(A^T)\text{.}\)
In Example 7.4.4, we found the singular value decomposition
Since there are two nonzero singular values, \(Rank(A) =2\) so that the reduced singular value decomposition is
Summary
This section has explored singular value decompositions, how to find them, and how they organize important information about a matrix.
- A singular value decomposition of a matrix \(A\) is a factorization where \(A=U\Sigma V^T\text{.}\) The matrix \(\Sigma\) has the same shape as \(A\text{,}\) and its only nonzero entries are the singular values of \(A\text{,}\) which appear in decreasing order on the diagonal. The matrices \(U\) and \(V\) are orthogonal and contain the left and right singular vectors.
- To find a singular value decomposition of a matrix, we construct the Gram matrix \(G=A^TA\text{,}\) which is symmetric. The singular values of \(A\) are the square roots of the eigenvalues of \(G\text{,}\) and the right singular vectors \(\mathbf v_j\) are the associated eigenvectors of \(G\text{.}\) The left singular vectors \(\mathbf u_j\) are determined from the relationship \(A\mathbf v_j=\sigma_j\mathbf u_j\text{.}\)
- A singular value decomposition organizes fundamental information about a matrix. For instance, the number of nonzero singular values is the rank \(r\) of the matrix. The first \(r\) left singular vectors form an orthonormal basis for \(Col(A)\) with the remaining left singular vectors forming an orthonormal basis of \(Nul(A^T)\text{.}\) The first \(r\) right singular vectors form an orthonormal basis for \(Col(A^T)\) while the remaining right singular vectors form an orthonormal basis of \(Nul(A)\text{.}\)
- If \(A\) is a rank \(r\) matrix, we can write a reduced singular value decomposition as \(A=U_r\Sigma_rV_r^T\) where the columns of \(U_r\) form an orthonormal basis for \(Col(A)\text{,}\) the columns of \(V_r\) form an orthonormal basis for \(Col(A^T)\text{,}\) and \(\Sigma_r\) is an \(r\times r\) diagonal, invertible matrix.
Exercises 7.4.5Exercises
Consider the matrix \(A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & -1 & 2 \\ \end{bmatrix} \text{.}\)
- Find the Gram matrix \(G=A^TA\) and use it to find the singular values and right singular vectors of \(A\text{.}\)
- Find the left singular vectors.
- Form the matrices \(U\text{,}\) \(\Sigma\text{,}\) and \(V\) and verify that \(A=U\Sigma V^T\text{.}\)
- What is \(Rank(A)\) and what does this say about \(Col(A)\text{?}\)
- Determine an orthonormal basis for \(Nul(A)\text{.}\)
Find singular value decompositions for the following matrices:
- \(\begin{bmatrix} 0 & 0 \\ 0 & -8 \end{bmatrix}\text{.}\)
- \(\begin{bmatrix} 2 & 3 \\ 0 & 2 \end{bmatrix}\text{.}\)
- \(\displaystyle \begin{bmatrix} 4 & 0 & 0 \\ 0 & 0 & 2 \end{bmatrix}\)
- \(\displaystyle \begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & 2 \end{bmatrix}\)
Consider the matrix \(A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \text{.}\)
- Find a singular value decomposition of \(A\) and verify that it is also an orthogonal diagonalization of \(A\text{.}\)
- If \(A\) is a symmetric, positive semidefinite matrix, explain why a singular value decomposition of \(A\) is an orthogonal diagonalization of \(A\text{.}\)
Suppose that the matrix \(A\) has the singular value decomposition
- What are the dimensions of \(A\text{?}\)
- What is \(Rank(A)\text{?}\)
- Find orthonormal bases for \(Col(A)\text{,}\) \(Nul(A)\text{,}\) \(Col(A^T)\text{,}\) and \(Nul(A^T)\text{.}\)
- Find the orthogonal projection of \(\mathbf b=\fourvec102{-1}\) onto \(Col(A)\text{.}\)
Consider the matrix \(A = \begin{bmatrix} 1 & 0 & -1 \\ 2 & 2 & 0 \\ -1 & 1 & 2\\ \end{bmatrix} \text{.}\)
- Construct the Gram matrix \(G\) and use it to find the singular values and right singular vectors \(\mathbf v_1\text{,}\) \(\mathbf v_2\text{,}\) and \(\mathbf v_3\) of \(A\text{.}\) What are the matrices \(\Sigma\) and \(V\) in a singular value decomposition?
- What is \(Rank(A)\text{?}\)
- Find as many left singular \(\mathbf u_j\) as you can using the relationship \(A\mathbf v_j=\sigma_j\mathbf u_j\text{.}\)
- Find an orthonormal basis for \(Nul(A^T)\) and use it to construct the matrix \(U\) so that \(A=U\Sigma V^T\text{.}\)
- State an orthonormal basis for \(Nul(A)\) and an orthonormal basis for \(Col(A)\text{.}\)
Consider the matrix \(B=\begin{bmatrix} 1 & 0 \\ 2 & -1 \\ 1 & 2 \end{bmatrix}\) and notice that \(B=A^T\) where \(A\) is the matrix in Exercise <IP_ADDRESS>.
- Use your result from Exercise <IP_ADDRESS> to find a singular value decomposition of \(B=U\Sigma V^T\text{.}\)
- What is \(Rank(B)\text{?}\) Determine a basis for \(Col(B)\) and \(Col(B)^\perp\text{.}\)
- Suppose that \(\mathbf b=\threevec{-3}47\text{.}\) Use the bases you found in the previous part of this exericse to write \(\mathbf b=\widehat{\mathbf{b}}+\mathbf b^\perp\text{,}\) where \(\widehat{\mathbf{b}}\) is in \(Col(B)\) and \(\mathbf b^\perp\) is in \(Col(B)^\perp\text{.}\)
- Find the least squares approximate solution to the equation \(B\mathbf x=\mathbf b\text{.}\)
Suppose that \(A\) is a square \(m\times m\) matrix with singular value decomposition \(A=U\Sigma V^T\text{.}\)
- If \(A\) is invertible, find a singular value decomposition of \(A^{-1}\text{.}\)
- What condition on the singular values must hold for \(A\) to be invertible?
- How are the singular values of \(A\) and the singular values of \(A^{-1}\) related to one another?
- How are the right and left singular vectors of \(A\) related to the right and left singular vectors of \(A^{-1}\text{?}\)
- If \(Q\) is an orthogonal matrix, remember that \(Q^TQ=I\text{.}\) Explain why \(\det Q = \pm 1\text{.}\)
- If \(A=U\Sigma V^T\) is a singular value decomposition of a square matrix \(A\text{,}\) explain why \(|\det A|\) is the product of the singular values of \(A\text{.}\)
- What does this say about the singular values of \(A\) if \(A\) is invertible?
If \(A\) is a matrix and \(G=A^TA\) its Gram matrix, remember that
- For a general matrix \(A\text{,}\) explain why the eigenvalues of \(G\) are nonnegative.
- Given a symmetric matrix \(A\) having an eigenvalue \(\lambda\text{,}\) explain why \(\lambda^2\) is an eigenvalue of \(G\text{.}\)
- If \(A\) is symmetric, explain why the singular values of \(A\) equal the absolute value of its eigenvalues: \(\sigma_j = |\lambda_j|\text{.}\)
Determine whether the following statements are true or false and explain your reasoning.
- If \(A=U\Sigma V^T\) is a singular value decomposition of \(A\text{,}\) then \(G=V(\Sigma^T\Sigma)V^T\) is an orthogonal diagonalization of its Gram matrix.
- If \(A=U\Sigma V^T\) is a singular value decomposition of a rank 2 matrix \(A\text{,}\) then \(\mathbf v_1\) and \(\mathbf v_2\) form an orthonormal basis for the column space \(Col(A)\text{.}\)
- If \(A\) is a diagonalizable matrix, then its set of singular values is the same as its set of eigenvalues.
- If \(A\) is a \(10\times7\) matrix and \(\sigma_7 = 4\text{,}\) then the columns of \(A\) are linearly independent.
- The Gram matrix is always orthogonally diagonalizable.
Suppose that \(A=U\Sigma V^T\) is a singular value decomposition of the \(m\times n\) matrix \(A\text{.}\) If \(\sigma_1,\ldots,\sigma_r\) are the nonzero singular values, the general form of the matrix \(\Sigma\) is
- If you know that the columns of \(A\) are linearly independent, what more can you say about the form of \(\Sigma\text{?}\)
- If you know that the columns of \(A\) span \(\mathbb R^m\text{,}\) what more can you say about the form of \(\Sigma\text{?}\)
- If you know that the columns of \(A\) are linearly independent and span \(\mathbb R^m\text{,}\) what more can you say about the form of \(\Sigma\text{?}\)


Introduction
Skills to Develop
- Understand some of the historically fundamental aspects of chemistry
Chemistry is different from physics. The chemical theories that we use to understand and predict aren't written in math. They also aren't based on anything we can see directly, like a ball falling. Instead, they are based in images, patterns, symmetry and imagination. How do we know what metaphors we can use to imagine, and predict accurately, things that we can't see? When you think about it, it's amazing how much chemists were able to figure out.
John Dalton proposed his atomic theory around 1805. He said:
- All matter is composed of atoms
- Atoms come in different types, called elements
- Atoms of each element have a distinct mass
- Each atom of a given element is identical to every other atom of that element
- Atoms are not created, destroyed or changed when chemical changes occur
Scientists scoffed, laughed at him, or at least remained skeptical, for at least 50 years after that, because how could he know? But he was almost completely right.
A little later, in 1830, Jons Jacob Berzelius proposed that even if you have the same composition (the same number of atoms of each element) you can have two different molecules (a combination of atoms), because the atoms are arranged differently. The first example discovered was silver fulminate (AgCNO, very explosive) and silver cyanate (AgOCN, a non-explosive grayish powder).
And then in 1874 Jacobus Henricus van't Hoff proposed that when carbon atoms form bonds to four other atoms, those other atoms around them have a particular arrangement in space: a tetrahedron . A critic said:
"A Dr. J. H. van't Hoff who is employed at the Veterinary School in Utrecht appears to find exact chemical research unsuited to his tastes. He finds it more suitable to mount Pegasus (obviously loaned from the Veterinary School) and to proclaim ... how, during his flight to the top of the chemical Parnassus, the atoms appeared to be arranged in the universe." (In ancient Greek mythology, Pegasus is a horse with wings, and Parnassus is a mountain associated with art and knowledge because the Muses were said to live there.)
You can tell from that how weird it seemed, at the time, to claim a particular spatial arrangement of tiny particles that nobody had ever seen. But van't Hoff, like Dalton and Berzelius, was right. How did they manage this? That will be one of the questions we answer in this class.
In fact, before the physicists had admitted that atoms exist, before they had been proven directly, chemists had already published many chemical structures showing how atoms were arranged in molecules.
Summary
Chemistry is the study of the composition and structure of chemical substances and the changes that they undergo. An element is one type of atom. An atom is the smallest particle of an element that retains all of the element's properties. A molecule is a combination of atoms of the same element. A substance's chemical composition involves having a piece of the substance being made of the same number of atoms as well as the the same type(s) of atoms.


5.8: The Chicago School
[1]
[2] Upon further investigation, Shaw and McKay noticed three qualitative differences in the transitional zone compared to other zones. First, the physical status included the invasion of industry and the largest number of condemned buildings. When many buildings are in disrepair, population levels decrease. Second, the population composition was also different. The zone in transition had higher concentrations of foreign-born and African-American heads of families. It also had a transient population. Third, the transitional zone had socioeconomic differences with the highest rates of welfare, lowest median rent, the lowest percentage of family-owned houses. Interrelated, the zone also had the highest rates of infant deaths, tuberculosis, and mental illness.
Social disorganization is the inability of social institutions to control an individual’s behavior. Since the zone in transition had people moving in and moving out at such high rates, social institutions (like family, school, religion, government, economy) and members could no longer agree on essential norms and values. As earlier stated, many residents were foreign. Thus, speaking different languages and having different religious beliefs may have prevented neighbors from talking to one another and solidifying community bonds. Overall, Shaw and McKay were two of the first theorists to put forth the premise that community characteristics matter when discussing criminal behavior.


20.7: Momentum and Radiation Pressure
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Learning Objectives
By the end of this section, you will be able to:
- Describe the relationship of the radiation pressure and the energy density of an electromagnetic wave
- Explain how the radiation pressure of light, while small, can produce observable astronomical effects
Material objects consist of charged particles. An electromagnetic wave incident on the object exerts forces on the charged particles, in accordance with the Lorentz force . These forces do work on the particles of the object, increasing its energy, as discussed in the previous section. The energy that sunlight carries is a familiar part of every warm sunny day. A much less familiar feature of electromagnetic radiation is the extremely weak pressure that electromagnetic radiation produces by exerting a force in the direction of the wave. This force occurs because electromagnetic waves contain and transport momentum.
To understand the direction of the force for a very specific case, consider a plane electromagnetic wave incident on a metal in which electron motion, as part of a current, is damped by the resistance of the metal, so that the average electron motion is in phase with the force causing it. This is comparable to an object moving against friction and stopping as soon as the force pushing it stops (Figure \(\PageIndex{1}\)). When the electric field is in the direction of the positive y -axis, electrons move in the negative y -direction, with the magnetic field in the direction of the positive z -axis. By applying the right-hand rule, and accounting for the negative charge of the electron, we can see that the force on the electron from the magnetic field is in the direction of the positive x -axis, which is the direction of wave propagation. When the \(\vec{E}\) field reverses, the \(\vec{B}\) field does too, and the force is again in the same direction. Maxwell’s equations together with the Lorentz force equation imply the existence of radiation pressure much more generally than this specific example, however.
Maxwell predicted that an electromagnetic wave carries momentum. An object absorbing an electromagnetic wave would experience a force in the direction of propagation of the wave. The force corresponds to radiation pressure exerted on the object by the wave. The force would be twice as great if the radiation were reflected rather than absorbed.
Maxwell’s prediction was confirmed in 1903 by Nichols and Hull by precisely measuring radiation pressures with a torsion balance. The schematic arrangement is shown in Figure \(\PageIndex{2}\). The mirrors suspended from a fiber were housed inside a glass container. Nichols and Hull were able to obtain a small measurable deflection of the mirrors from shining light on one of them. From the measured deflection, they could calculate the unbalanced force on the mirror, and obtained agreement with the predicted value of the force.
The radiation pressure \(p_{rad}\) applied by an electromagnetic wave on a perfectly absorbing surface turns out to be equal to the energy density of the wave:
\[ \underbrace{p_{rad} = u \space} _{ \text{Perfect absorber}}. \label{eq5} \]
If the material is perfectly reflecting, such as a metal surface, and if the incidence is along the normal to the surface, then the pressure exerted is twice as much because the momentum direction reverses upon reflection:
\[ \underbrace{ p_{rad} = 2u }_{ \text{Perfect reflector}}. \label{eq10} \]
We can confirm that the units are right:
\[[u] = \dfrac{J}{m^3} = \dfrac{N \cdot m}{m^3} = \dfrac{N}{m^2} = units \, of \, pressure. \nonumber \]
Equations \ref{eq5} and \ref{eq10} give the instantaneous pressure, but because the energy density oscillates rapidly, we are usually interested in the time-averaged radiation pressure, which can be written in terms of intensity:
\[ p = \langle p_{rad}\rangle = \begin{cases}
I/c & \text{Perfect absorber} \\
2I/c & \text{Perfect reflector}
\end{cases}
\label{eq20} \]
Radiation pressure plays a role in explaining many observed astronomical phenomena, including the appearance of comets . Comets are basically chunks of icy material in which frozen gases and particles of rock and dust are embedded. When a comet approaches the Sun, it warms up and its surface begins to evaporate. The coma of the comet is the hazy area around it from the gases and dust. Some of the gases and dust form tails when they leave the comet. Notice in Figure \(\PageIndex{3}\) that a comet has two tails. The ion tail (or gas tail ) is composed mainly of ionized gases. These ions interact electromagnetically with the solar wind, which is a continuous stream of charged particles emitted by the Sun. The force of the solar wind on the ionized gases is strong enough that the ion tail almost always points directly away from the Sun. The second tail is composed of dust particles. Because the dust tail is electrically neutral, it does not interact with the solar wind. However, this tail is affected by the radiation pressure produced by the light from the Sun. Although quite small, this pressure is strong enough to cause the dust tail to be displaced from the path of the comet.
Example \(\PageIndex{1}\): Halley’s Comet
On February 9, 1986, Comet Halley was at its closest point to the Sun, about \(9.0 \times 10^{10} m\) from the center of the Sun. The average power output of the Sun is \(3.8 \times 10^{26} \, W\).
- Calculate the radiation pressure on the comet at this point in its orbit. Assume that the comet reflects all the incident light.
- Suppose that a 10-kg chunk of material of cross-sectional area \(4.0 \times 10^{-2} m^2\) breaks loose from the comet. Calculate the force on this chunk due to the solar radiation. Compare this force with the gravitational force of the Sun.
Strategy
Calculate the intensity of solar radiation at the given distance from the Sun and use that to calculate the radiation pressure. From the pressure and area, calculate the force.
Solution
a. The intensity of the solar radiation is the average solar power per unit area. Hence, at \(9.0 \times 10^{10} m\) from the center of the Sun, we have
\[\begin{align} I &= S_{avg} \nonumber \\[4pt] &= \dfrac{3.8 \times 10^{26} \, W}{4\pi (9.0 \times 10^{10} \, m)^2} \nonumber \\[4pt] &= 3.7 \times 10^3 \, W/m^2. \nonumber \end{align} \nonumber \]
Assuming the comet reflects all the incident radiation, we obtain from Equation \ref{eq20}
\[\begin{align}p &= \dfrac{2I}{c} \nonumber \\[4pt] &= \dfrac{2(3.7 \times 10^3 \, W/m^2)}{3.00 \times 10^8 \, m/s} \nonumber \\[4pt] &= 2.5 \times 10^{-5} \, N/m^2. \nonumber \end{align} \nonumber \]
b. The force on the chunk due to the radiation is
\[\begin{align}F &= pA \nonumber \\[4pt] &= (2.5 \times 10^{-5} N/m^2)(4.0 \times 10^{-2} m^2) \nonumber \\[4pt] &= 1.0 \times 10^{-6} \, N, \nonumber \end{align} \nonumber \]
whereas the gravitational force of the Sun is
\[\begin{align} F_g &= \dfrac{GMm}{r^2} \nonumber \\[4pt] &= \dfrac{(6.67 \times 10^{-11} \, N \cdot m^2 /kg^2)(2.0 \times 10^{30} kg)(10 \, kg)}{(9.0 \times 10^{10} m)^2} \nonumber \\[4pt] &= 0.16 \, N. \nonumber \end{align} \nonumber \]
Significance
The gravitational force of the Sun on the chunk is therefore much greater than the force of the radiation.
After Maxwell showed that light carried momentum as well as energy, a novel idea eventually emerged, initially only as science fiction. Perhaps a spacecraft with a large reflecting light sail could use radiation pressure for propulsion. Such a vehicle would not have to carry fuel. It would experience a constant but small force from solar radiation, instead of the short bursts from rocket propulsion. It would accelerate slowly, but by being accelerated continuously, it would eventually reach great speeds. A spacecraft with small total mass and a sail with a large area would be necessary to obtain a usable acceleration.
When the space program began in the 1960s, the idea started to receive serious attention from NASA. The most recent development in light propelled spacecraft has come from a citizen-funded group, the Planetary Society. It is currently testing the use of light sails to propel a small vehicle built from CubeSats , tiny satellites that NASA places in orbit for various research projects during space launches intended mainly for other purposes.
The LightSail spacecraft shown below (Figure \(\PageIndex{4}\)) consists of three CubeSats bundled together. It has a total mass of only about 5 kg and is about the size as a loaf of bread. Its sails are made of very thin Mylar and open after launch to have a surface area of \(32 \, m^2\).
Example \(\PageIndex{2}\): LightSail Acceleration
The first LightSail spacecraft was launched in 2015 to test the sail deployment system. It was placed in low-earth orbit in 2015 by hitching a ride on an Atlas 5 rocket launched for an unrelated mission. The test was successful, but the low-earth orbit allowed too much drag on the spacecraft to accelerate it by sunlight. Eventually, it burned in the atmosphere, as expected. The next Planetary Society’s LightSail solar sailing spacecraft is scheduled for 2018.
The Lightsail is based on the on NASA's NanoSail-D project. (Public domain; NASA).
LightSail Acceleration
The intensity of energy from sunlight at a distance of 1 AU from the Sun is \(1370 \, W/m^2\). The LightSail spacecraft has sails with total area of \(32 \, m^2\) and a total mass of 5.0 kg. Calculate the maximum acceleration LightSail spacecraft could achieve from radiation pressure when it is about 1 AU from the Sun.
Strategy
The maximum acceleration can be expected when the sail is opened directly facing the Sun. Use the light intensity to calculate the radiation pressure and from it, the force on the sails. Then use Newton’s second law to calculate the acceleration.
Solution
The radiation pressure is
\[F = pA = 2uA = \dfrac{2I}{c}A = \dfrac{2(1370 \, W/m^2)(32 \, m^2)}{(3.00 \times 10^8 m/s)} = 2.92 \times 10^{-4} N. \nonumber \]
The resulting acceleration is
\[a = \dfrac{F}{m} = \dfrac{2.92 \times 10^{-4} N}{5.0 \, kg} = 5.8 \times 10^{-5} m/s^2. \nonumber \]
Significance
If this small acceleration continued for a year, the craft would attain a speed of 1829 m/s, or 6600 km/h.
Exercise \(\PageIndex{1}\)
How would the speed and acceleration of a radiation-propelled spacecraft be affected as it moved farther from the Sun on an interplanetary space flight?
Solution
Its acceleration would decrease because the radiation force is proportional to the intensity of light from the Sun, which decreases with distance. Its speed, however, would not change except for the effects of gravity from the Sun and planets.


2.3: BET Surface Area Analysis of Nanoparticles
Introduction
In the past few years, nanotechnology research has expanded out of the chemistry department and into the fields of medicine, energy, aerospace and even computing and information technology. With bulk materials, the surface area to volume is insignificant in relation to the number of atoms in the bulk, however when the particles are only 1 to 100 nm across, different properties begin to arise. For example, commercial grade zinc oxide has a surface area range of 2.5 to 12 m 2 /g while nanoparticle zinc oxide can have surface areas as high as 54 m 2 /g . The nanoparticles have superior UV blocking properties when compared to the bulk material, making them useful in applications such as sunscreen. Many useful properties of nanoparticles rise from their small size, making it very important to be able to determine their surface area.
Overview of BET Theory
The BET theory was developed by Stephen Brunauer (Figure \(\PageIndex{1}\) ), Paul Emmett (Figure \(\PageIndex{2}\) ), and Edward Teller (Figure \(\PageIndex{3}\) ) in 1938. The first letter of each publisher’s surname was taken to name this theory. The BET theory was an extension of the Langmuir theory, developed by Irving Langmuir (Figure \(\PageIndex{4}\) ) in 1916.
The Langmuir theory relates the monolayer adsorption of gas molecules (Figure \(\PageIndex{5}\) ), also called adsorbates, onto a solid surface to the gas pressure of a medium above the solid surface at a fixed temperature to \ref{1} , where θ is the fractional cover of the surface, P is the gas pressure and α is a constant.
\[ \Theta \ =\ \frac{\alpha \cdot P}{1\ +\ (\alpha \cdot P)} \label{1} \]
The Langmuir theory is based on the following assumptions:
- All surface sites have the same adsorption energy for the adsorbate, which is usually argon, krypton or nitrogen gas. The surface site is defined as the area on the sample where one molecule can adsorb onto.
- Adsorption of the solvent at one site occurs independently of adsorption at neighboring sites.
- Activity of adsorbate is directly proportional to its concentration.
- Adsorbates form a monolayer.
- Each active site can be occupied only by one particle.
The Langmuir theory has a few flaws that are addressed by the BET theory. The BET theory extends the Langmuir theory to multilayer adsorption (Figure \(\PageIndex{1}\) ) with three additional assumptions:
- Gas molecules will physically adsorb on a solid in layers infinitely.
- The different adsorption layers do not interact.
- The theory can be applied to each layer.
How does BET Work?
Adsorption is defined as the adhesion of atoms or molecules of gas to a surface. It should be noted that adsorption is not confused with absorption, in which a fluid permeates a liquid or solid. The amount of gas adsorbed depends on the exposed surface are but also on the temperature, gas pressure and strength of interaction between the gas and solid. In BET surface area analysis, nitrogen is usually used because of its availability in high purity and its strong interaction with most solids. Because the interaction between gaseous and solid phases is usually weak, the surface is cooled using liquid N 2 to obtain detectable amounts of adsorption. Known amounts of nitrogen gas are then released stepwise into the sample cell. Relative pressures less than atmospheric pressure is achieved by creating conditions of partial vacuum. After the saturation pressure, no more adsorption occurs regardless of any further increase in pressure. Highly precise and accurate pressure transducers monitor the pressure changes due to the adsorption process. After the adsorption layers are formed, the sample is removed from the nitrogen atmosphere and heated to cause the adsorbed nitrogen to be released from the material and quantified. The data collected is displayed in the form of a BET isotherm, which plots the amount of gas adsorbed as a function of the relative pressure. There are five types of adsorption isotherms possible.
Type I Isotherm
Type I is a pseudo-Langmuir isotherm because it depicts monolayer adsorption (Figure \(\PageIndex{6}\) ). A type I isotherm is obtained when P/P o < 1 and c > 1 in the BET equation, where P/P o is the partial pressure value and c is the BET constant, which is related to the adsorption energy of the first monolayer and varies from solid to solid. The characterization of microporous materials, those with pore diameters less than 2 nm, gives this type of isotherm.
Type II Isotherm
A type II isotherm (Figure \(\PageIndex{7}\) ) is very different than the Langmuir model. The flatter region in the middle represents the formation of a monolayer. A type II isotherm is obtained when c > 1 in the BET equation. This is the most common isotherm obtained when using the BET technique. At very low pressures, the micropores fill with nitrogen gas. At the knee, monolayer formation is beginning and multilayer formation occurs at medium pressure. At the higher pressures, capillary condensation occurs.
Type III Isotherm
A type III isotherm (Figure \(\PageIndex{8}\) ) is obtained when the c < 1 and shows the formation of a multilayer. Because there is no asymptote in the curve, no monolayer is formed and BET is not applicable.
Type IV Isotherm
Type IV isotherms (Figure \(\PageIndex{9}\) ) occur when capillary condensation occurs. Gases condense in the tiny capillary pores of the solid at pressures below the saturation pressure of the gas. At the lower pressure regions, it shows the formation of a monolayer followed by a formation of multilayers. BET surface area characterization of mesoporous materials, which are materials with pore diameters between 2 - 50 nm, gives this type of isotherm.
Type V Isotherm
Type V isotherms (Figure \(\PageIndex{10}\) ) are very similar to type IV isotherms and are not applicable to BET.
Calculations
The BET Equation, \ref{2} , uses the information from the isotherm to determine the surface area of the sample, where X is the weight of nitrogen adsorbed at a given relative pressure (P/Po), X m is monolayer capacity, which is the volume of gas adsorbed at standard temperature and pressure (STP), and C is constant. STP is defined as 273 K and 1 atm.
\[ \frac{1}{X[(P_{0}/P)-1]} = \frac{1}{X_{m}C} + \frac{C-1}{X_{m}C} (\frac{P}{P_{0}}) \label{2} \]
Multi-point BET
Ideally five data points, with a minimum of three data points, in the P/P 0 range 0.025 to 0.30 should be used to successfully determine the surface area using the BET equation. At relative pressures higher than 0.5, there is the onset of capillary condensation, and at relative pressures that are too low, only monolayer formation is occurring. When the BET equation is plotted, the graph should be of linear with a positive slope. If such a graph is not obtained, then the BET method was insufficient in obtaining the surface area.
- The slope and y-intercept can be obtained using least squares regression.
- The monolayer capacity X m can be calculated with \ref{3} .
- Once X m is determined, the total surface area S t can be calculated with the following equation, where L av is Avogadro’s number, A m is the cross sectional area of the adsorbate and equals 0.162 nm 2 for an absorbed nitrogen molecule, and M v is the molar volume and equals 22414 mL, \ref{4} .
\[ X_{m}\ = \frac{1}{s\ +\ i} = \frac{C-1}{C_{s}} \label{3} \]
\[S\ = \frac{X_{m} L_{av} A_{m}}{M_{v}} \label{4} \]
Single point BET can also be used by setting the intercept to 0 and ignoring the value of C. The data point at the relative pressure of 0.3 will match up the best with a multipoint BET. Single point BET can be used over the more accurate multipoint BET to determine the appropriate relative pressure range for multi-point BET.
Sample Preparation and Experimental Setup
Prior to any measurement the sample must be degassed to remove water and other contaminants before the surface area can be accurately measured. Samples are degassed in a vacuum at high temperatures. The highest temperature possible that will not damage the sample’s structure is usually chosen in order to shorten the degassing time. IUPAC recommends that samples be degassed for at least 16 hours to ensure that unwanted vapors and gases are removed from the surface of the sample. Generally, samples that can withstand higher temperatures without structural changes have smaller degassing times. A minimum of 0.5 g of sample is required for the BET to successfully determine the surface area.
Samples are placed in glass cells to be degassed and analyzed by the BET machine. Glass rods are placed within the cell to minimize the dead space in the cell. Sample cells typically come in sizes of 6, 9 and 12 mm and come in different shapes. 6 mm cells are usually used for fine powders, 9 mm cells for larger particles and small pellets and 12 mm are used for large pieces that cannot be further reduced. The cells are placed into heating mantles and connected to the outgas port of the machine.
After the sample is degassed, the cell is moved to the analysis port (Figure \(\PageIndex{11}\) ). Dewars of liquid nitrogen are used to cool the sample and maintain it at a constant temperature. A low temperature must be maintained so that the interaction between the gas molecules and the surface of the sample will be strong enough for measurable amounts of adsorption to occur. The adsorbate, nitrogen gas in this case, is injected into the sample cell with a calibrated piston. The dead volume in the sample cell must be calibrated before and after each measurement. To do that, helium gas is used for a blank run, because helium does not adsorb onto the sample.
Shortcomings of BET
The BET technique has some disadvantages when compared to NMR, which can also be used to measure the surface area of nanoparticles. BET measurements can only be used to determine the surface area of dry powders. This technique requires a lot of time for the adsorption of gas molecules to occur. A lot of manual preparation is required.
The Surface Area Determination of Metal-Organic Frameworks
The BET technique was used to determine the surface areas of metal-organic frameworks (MOFs), which are crystalline compounds of metal ions coordinated to organic molecules. Possible applications of MOFs, which are porous, include gas purification and catalysis. An isoreticular MOF (IRMOF) with the chemical formula Zn 4 O(pyrene-1,2-dicarboxylate) 3 (Figure \(\PageIndex{12}\) ) was used as an example to see if BET could accurately determine the surface area of microporous materials. The predicted surface area was calculated directly from the geometry of the crystals and agreed with the data obtained from the BET isotherms. Data was collected at a constant temperature of 77 K and a type II isotherm (Figure \(\PageIndex{13}\) ) was obtained.
The isotherm data obtained from partial pressure range of 0.05 to 0.3 is plugged into the BET equation, \ref{2} , to obtain the BET plot (Figure \(\PageIndex{14}\) ).
Using \ref{5} , the monolayer capactiy is determined to be 391.2 cm 3 /g.
\[ X_{m}\ = \frac{1}{(2.66E\ -\ 3)\ +\ (-5.212E\ -\ 0.05)} \label{5} \]
Now that X m is known, then \ref{6} can be used to determine that the surface area is 1702.3 m 2 /g.
\[S\ =\frac{391.2cm^{2} \ast 0.162nm^{2} \ast 6.02\ast 10^{23}}{22.414:L} \label{6} \]


15.9: Hearing
Animals use the sense of hearing for many different purposes. It is used to sense danger and enemies, to detect prey, to identify prospective mates and to communicate within social groups. Some animals (e.g. most bats and dolphins) use sound to “see” by echolocation. By sending out a cry and interpreting the echo, they sense obstacles or potential prey.
Structure of the Ear
Most of the ear, the organ of hearing, is hidden from view within the boney skull. It consists of three main regions: the outer ear, the middle ear and the inner ear (see diagram 15.9).
Diagram 15.9 - The ear
The outer ear consists of an ear canal leading inwards to a thin membrane known as the eardrum or tympanic membrane that stretches across the canal. Many animals have an external ear flap or pinna to collect and funnel the sound into the ear canal. The pinnae (plural of pinna) usually face forwards on the head but many animals can swivel them towards the source of the sound.
In dogs the ear canal is long and bent and often traps wax or provides an ideal habitat for mites, yeast and bacteria.
The middle ear consists of a cavity in the skull that is connected to the pharynx (throat) by a long narrow tube called the Eustachian tube . This links the middle ear to the outside air so that the air pressure on both sides of the eardrum can be kept the same. Everyone knows the uncomfortable feeling (and affected hearing) that occurs when you drive down a steep hill and the unequal air pressures on the two sides of the eardrum cause it to distort. The discomfort is relieved when you swallow because the Eustachian tubes open and the pressure on either side equalises.
Within the cavity of the middle ear are three of the smallest bones in the body, the auditory ossicles . They are known as the hammer, the anvil and the stirrup because of their resemblance to the shape of these objects. These tiny bones articulate (move against) each other and transfer the vibrations of the eardrum to the membrane covering the opening to the inner ear.
The inner ear is a complicated series of fluid-filled tubes imbedded in the bone of the skull. It consists of two main parts. These are the cochlea where sound waves are converted to nerve impulses and the vestibular organ that is associated with the sense of balance and has no role in hearing (see later).
The cochlea looks rather like a coiled up snail shell. Within it there are specialised cells with fine hairs on their surface that respond to the movement of the fluid within the cochlea by producing nervous impulses that travel to the brain along the auditory nerve .
How The Ear Hears
Sound waves can be thought of as vibrations in the air. They are collected by the ear pinna and pass down the ear canal where they cause the eardrum to vibrate. (An interesting fact is that when you are listening to someone speaking your eardrum vibrates at exactly the same rate as the vocal cords of the person speaking to you).
The vibration of the eardrum sets the three tiny bones in the middle ear moving against each other so that the vibration is transferred to the membrane covering the opening to the inner ear. As well as transferring the vibration, the tiny ear bones also amplify it. The three tiny bones are called the stirrup, anvil, and hammer. They were called such of their form. In the human ear this amplification is about 20 times while in desert-dwelling animals like the kangaroo rat it is 100 times. This acute hearing warns them of the approach of predators like owls and snakes, even in the dark.
The vibration causes waves in the fluid in the inner ear that pass down the cochlea. These waves stimulate the tiny hair cells to produce nerve impulses that travel via the auditory nerve to the cerebral cortex of the brain where they are interpreted as sound.
To summarise: The route sound waves take as they pass through the ear is: External ear | tympanic membrane | ear ossicles | inner ear |cochlear | hair cells
The hair cells generate a nerve impulse that travels down the auditory nerve to the brain.
Remember that sound waves do not pass along the Eustachian tube. Its function is to equalise the air pressure on either side of the tympanic membrane.


6.16: U.S. Census Bureau and Demographic Data
- List the types of demographic data a retailer can obtain from the U.S. census bureau
So remember back to the last section! We found this awesome spot that would be ideal for our new retail location. Well, we think it will be awesome, but there are so many things to think about and analyze before we can say for sure this space will work! We need to know if there are enough people, with a high enough income and the right demographic to shop in our retail store. We also need to make sure we have parking and access, along with knowing the competitive environment. So where should we start? Well, the U.S. Census Bureau collects tons of information!!
The U.S. Census Bureau
A census gathers information from every household in the United States every 10 years. This data includes:
- Number of people in the household.
- Household relationships
- Age, sex, race and marital status
- Number of household units in each address
- Status of the plumbing facilities, number of rooms.
- Owner occupied or rental
- The value of the home and the vacancy status
Census data can be out of date, since it is only gathered every 10 years, but it is a good place to start your research once you have chosen a general area.
This Quickstart will help you pinpoint the demographics you need to get started! Note that this Quickstart is only available for towns over 5,000 people, so if you are working with a small town, you may need to do some different research!
Let’s look at the Census QuickFacts about Wausau, Wisconsin as an example. The drop down lets you select among a variety of information choices.
So, looking at the median income of the Wausau, Wisconsin area, if you are planning to put in an upscale steakhouse, do you think it would make sense? What if you were planning a McDonald’s or a Taco Bell?
What other information might you be interested in reviewing?
In this Quickstart, you have access to a variety of topics such as population, age and sex, race, health and economy, among others. Take some time and do a little research on the U.S. Census website. Think about a business location and business idea. Then wander out there and do some research.
Do you think the community you chose would support your business idea? Is the population enough? Does the education level support your business?
So remember, this data is only collected every 10 years, so it may be a bit outdated. It is a good starting point though to review your idea to see if the location is feasible.
Contributors and Attributions
- U.S. Census Bureau and Demographic Data. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution


2.6: Virtue Ethics
Virtue ethics has its historical background in ancient Greece and was primarily developed by Aristotle. For the purposes of law enforcement, the major foundation in virtue ethics is the idea that if you are a good person, you will do good things, and to be good, you must do good (Pollock, 2007). In essence, we do not do good things because of an analysis of the end result or of an equation to decide how many people to help versus harm. Rather, we do the right thing, or good thing, because of our good character as demonstrated throughout our life. Therefore, the good act is an automatic response requiring little thought. However, when faced with complex ethical dilemmas, the person who has demonstrated a life of good character will show good character, using temperance and intellect. The real question for Aristotle was not, “what should I do?” but rather “what type of person ought I be?” When our answer is that we ought to be a virtuous person, we are likely to act in a virtuous manner, and therefore in an ethical manner.
Aristotle also spoke of flourishing in life, or living in a state of well-being. He used the word eudaimonia (from the Greek eu, “ good,” and daimon, “spirit;” commonly translated as “happiness” or “welfare,” but more accurately as “human flourishing”) to express the state of well-being and living a flourishing life. Within this context, Aristotle concentrated on virtues and vices. Virtues are strengths of a person’s character that promote flourishing and well-being (Hinman, 2013). Conversely, vices are character flaws that impede flourishing and limit one’s sense of well-being.
Hinman (2013) writes of different types of virtues that Aristotle proposed:
- Executive virtues are examples of “strength of will,” such as courage and perseverance.
- Moral virtues are related to moral goodness. Examples are compassion, generosity, truthfulness, and good temper.
- Intellectual virtues are related to the ability to consider options. Examples are wittiness, wisdom, and understanding.
When we look at some of these virtues collectively, we can see that they project attributes that we want law enforcement personnel to possess. In a law enforcement context, society has expectations of officers who:
- Are courageous . Officers who are willing to put themselves in harm’s way, in order to enforce the law, to protect people and property and to prevent crime.
- Demonstrate perseverance . Officers who are not easily deterred from doing the right thing or investigating crimes.
- Exhibit compassion. Officers who are able to empathize and sympathize with lawbreakers and victims and who understand that situations are complex and that everyone deserves respect.
- Act with generosity. Officers who offer themselves off duty by volunteering and who try to better the lives of others through community service.
- Show truthfulness. Officers who are trustworthy and who can be counted on to speak the truth, even when the truth is embarrassing, or results in a not-guilty decision in a case that is important to the officer.
- Display good temper. Officers who, when confronted with difficult situations, stay calm and who are able to withstand pressure to react physically or verbally.
The virtues listed above are attractive to law enforcement agencies, and people who demonstrate these virtues are those who law enforcement agencies and all other branches of public service want. Vichio suggests a list of core virtues that law enforcement personnel should possess (Fitch, 2014). They include:
- Prudence. Officers with the ability to decide the correct action to take when rules and policy are not present.
- Trust. Officers with the ability to be relied upon for truth. This must exist between officers and civilians, officers themselves, and officers and the courts.
- Effacement of self-interests. Officers who do not abuse their position of authority or gain favouritisms due to their position.
- Courage. Officers who place themselves in danger intellectually and physically. Officers who are not afraid of testifying in court and/or making arrests in tense and intimidating settings.
- Intellectual honesty. Officers who act while weighing what they learned in training and whose actions reflect their training and their academic abilities.
- Justice. Officers who treat everyone fairly, regardless of personal biases, and who act toward individuals as if looking through a veil of neutrality.
- Responsibility. Officers who understand what is right and that there are other courses of actions, but have the intent to do right. Officers who can be counted upon to keep oaths, and to be accountable.
The Center for American and International Law identifies what they term the Six Pillars of Character. They created these pillars with the assistance of 30 national leaders and ethicists. The six pillars that they identified as being the most important characteristics of an ethical police officer are:
- Trustworthiness. Includes integrity, promise-keeping, and loyalty.
- Respect . Treating everyone with respect, regardless of any biases or provocations.
- Responsibility . Includes accountability, pursuit of excellence, and self-restraint.
- Justice and fairness . Includes equity and demonstrating due process.
- Caring . Showing concern for others. Showing consideration for decisions that affect others.
- Civic virtue and citizenship . Being socially conscious. Demonstrating concern for one’s community.
Q. How can virtue ethics assist law enforcement in moral dilemmas?
As mentioned previously, law enforcement agencies place a great emphasis on good behaviour of their officers. One way to ensure a strong likelihood of good behaviour is to hire those who have moral character that reflect the values of the agency. In clearly identifying these characteristics, agencies are likely to attract those who also identify with these characteristics.
- Virtue ethics, at its core, is also simplistic, having two tenets that are important for law enforcement. There is no need to measure consequences or the morality of the action. Simply, the task is to be good and do good acts. If officers are good, they will act in a virtuous manner.
- There is a need to practise virtue. By practising being virtuous, you will become virtuous in difficult situations automatically. Given this view, it is critical for law enforcement agencies to ensure that applicants wanting to join the agency have practised being virtuous to the point where it has become a habit. Applicants who have practised the virtues listed above will be officers who demonstrate those virtues by habit.


4.3: Work Behaviours
Assertive Behaviour
When you are assertive, you express your feelings, thoughts, and wishes clearly and directly. You stand up for your own rights without infringing on the rights of other people. You disagree when you think it is necessary. You are willing to modify your own behaviour in order to allow others to meet their needs. When you are treated badly by others, you are willing to set limits on the behaviour you will permit. An assertive person is cooperative and believes that compromise and communication will help all to reach their goals. Being assertive does not mean being aggressive.
Assertive behaviour is based on a set of rights and responsibilities that is a little different from what you might have learned within your family. For example, many people believe that it is selfish to put your needs before someone else’s. They may believe that it is shameful to make mistakes or admit to them. In contrast, assertive behaviour is based on a belief that you are an adult with these alternatives and legitimate rights:
- To act in your own interest
- To respect and stand up for yourself
- To express ideas, beliefs, and honest emotions
- To ask for what you want
- To be treated with respect and dignity
- To make mistakes
Of course, rights are also balanced with responsibilities. Assertive individuals:
- Respect and honour these rights for others
- Take responsibility for their own behaviour
- Take responsibility for their own mistakes
Components of Assertive Behaviour
Being assertive involves more than just what you say. It includes a set of non-verbal behaviours and a set of feelings about oneself.
Eye contact
Looking directly at a person when you are speaking shows that you are sincere, that you are interested in the conversation, and that you are sure of what you are saying. Looking away or down, or shifting your eyes constantly may convey the message that you lack confidence or are insincere. Staring or glaring at the person may indicate aggression.
Posture
An assertive individual stands or sits with an erect posture and maintains an appropriate distance from the person being addressed. Slumped posture, a hanging head, and moving away from the person being addressed may be interpreted as non-assertive. Moving in too close or leaning over the person may convey aggression.
Gestures
Threatening gestures such as a pointed finger or a clenched fist may intimidate the person being addressed. Conversely, gestures such as wringing hands, biting nails, or other nervous mannerisms may create the impression of passivity. When you are assertive, you either leave your hands by your sides or use appropriate gestures.
Facial expression and voice
A pleasant, steady, and strong speaking voice accompanied by facial expressions that are appropriate to the verbal message tells the other person that you mean what you say. A quivery, faltering voice that can barely be heard and an inappropriate expression may convey a message that will not be taken seriously even if the verbal part of the message is assertive.
“I” messages and observations
Assertive persons use “I” statements that indicate what they feel, think, and want. These statements begin with the word “I.” They include clear, direct observations about specific incidents and events. They do not make sweeping generalizations or vague, indirect statements that force the other person to guess the problem. They do not mislabel their inferences and conclusions about the behaviour of others as facts. They avoid laying blame or making judgments about the behaviour of others.
For example, James has come into work for three nights in a row and discovered that the kitchen was not thoroughly cleaned at the end of the previous shift. The next night, he makes a point of coming in a little early so that he can talk to the person who works on the previous shift.
Here are three possible approaches he could take:
- Aggressive : (shouting) “What an inconsiderate slob you are! The kitchen is always a pigsty when I come in to work.”
- Assertive : (calm) “I noticed the stove and countertops were not clean when I came in for work the last three nights. I felt very angry because I make a point of leaving the kitchen spotless each night.”
- Passive : (whining) “Is there some problem going on during the day that I should know about?”
Notice that the first approach is bound to lead to conflict, with someone winning the fight and the other person losing. It blames the other person for leaving the kitchen untidy, and assumes that the person is inconsiderate and a slob. It ignores the possibility that the restaurant was extraordinarily busy at lunch, that the kitchen was understaffed, or some other reasonable explanation.
The assertive approach is objective and informative. It allows James to express his feelings about what happened. It is more likely to get a positive response than the aggressive approach.
The passive approach (see below) forces the day staff to figure out what the problem is. James is not being clear and direct. By the time the problem is identified, everyone might be frustrated and angry.
One common approach that does not appear to fit into the above categories is to complain about the behaviour to others but not address the situation directly. For example, James might complain to the other staff working the evening shift. This may defuse some of his anger about the condition of the kitchen, but it does not solve the problem. It is also a passive approach. Over time, this approach leads to a build-up of hostility between the staff on the different shifts. Staff may indulge in various petty acts to get back at the other shift for their perceived shortcomings. In the end, the customer suffers. An active, assertive approach to solving conflict works best for everyone. It can be awkward to do but in the end it is the best approach for all parties involved.
Aggressive Behaviour
In contrast, some people behave aggressively. They pay little attention to the rights and needs of others. They achieve their goals at the expense of others. In communication, they may shout, threaten, and bluster to get their own way. They may humiliate other people by their communication and actions. They are highly competitive in their behaviour and believe that they can only win if others lose.
There are times when it is appropriate to be aggressive. For example, when your life is being threatened or when faced with an emergency, it is important to react swiftly and firmly. There is no time to think about consulting others and meeting their needs.
Passive or Non-Assertive Behaviour
Passive or non-assertive persons are not in control of their own destiny. They do not stand up for their own rights and may be taken advantage of by others. By allowing others to choose for them, they give up the opportunity to reach their own goals. In communication, non-assertive persons are often inhibited and indirect. They do not say what they feel or need, although they may try to express these needs indirectly. They often accommodate others or give in to others. They may feel frustrated, unhappy, or hurt in the process.
There may be times when it is appropriate to be passive. For example, if you have made a serious and costly error that could cost you your job, you should probably be prepared to hear out your supervisor without being assertive or aggressive.
No one is consistently assertive. You may be assertive with your co-workers, passive or non-assertive with your boss, and aggressive with your youngest brother or sister. Often people find it very difficult to be assertive when they are dealing with a stranger or someone in a position of authority. By learning and practising assertive behaviour, you can expand the number of situations in which you respond assertively. You can choose when to act assertively and when not to.


15.17: Net Present Value
Learning Outcomes
- Describe the net present value method
Net present value is defined as the difference between the present value of cash inflows and the present value of cash outflows over a period of time. It is used in capital budgeting to determine the profitability of a potential investment or project.
A positive net present value is a good thing, and denotes a project that will exceed the anticipated costs, both defined in present dollars. If an investment shows a negative present value, the company would lose money on the investment or project and should not proceed. So with this rule in hand, it would make sense to only follow through with projects with a positive net present value, or those that would make the company more money that the initial investment. It wouldn’t make sense to invest money that would not be made back, plus a profit, right?
Using this method can be challenging, as there are many ways to determine the value of future cash flows. As an example, if we are using the savings in labor as a part of the positive cash flow for an equipment purchase, if wages go up substantially due to economic change, it could create a much better positive cash flow. But if over the same time, economic pressure brings down the labor rate, it could have the opposite effect on the potential positive cash flow of the same purchase.
Example
A restaurant is considering the purchase of an additional location in a neighboring town. Looking at potential revenues of the additional location for the next five years, discounted to present value—let’s say comes to $250,000. If the owner of the additional location is willing to sell the restaurant for $250,000 or less, the purchase makes sense. If they are not willing to sell for less than $250,000 we would pass on this purchase. The awesome thing would be if the owner offered to sell for $100,000, creating a gain on the investment called intrinsic value.
There are pitfalls to the net present value approach, as there are will any of the methods. In our restaurant example, what if the current owner is selling due to impending competition moving in, or maybe a new highway is coming through that will divert traffic? Also, the payback period is only the five years as noted. Once our investment is earned back, will we continue to make enough money to keep the space open profitably? So many questions when we are making these capital budgeting decisions!
PRactice Questions
Learn More
For more practice check out this example on Net Present Value Method .


 It is a major cause of lower respiratory tract infections and hospital visits during infancy and childhood. A prophylactic medication (not a vaccine) exists for preterm-birth (under 35 weeks gestation) infants, and for infants with a congenital heart defect or bronchopulmonary dysplasia. Of those infected with RSV, 2–3% will develop bronchiolitis, necessitating hospitalization.
RSV is a negative-sense, single-stranded RNA virus of the family Paramyxoviridae, which includes common respiratory viruses such as those causing measles and mumps. RSV is a member of the paramyxovirus subfamily Pneumovirinae. RSV has ten genes encoding 11 proteins. There are two open reading frames of M2. NS1 and NS2 inhibit type I interferon activity. N encodes the nucleocapsid protein that associates with the genomic RNA forming the nucleocapsid. M encodes the matrix protein required for viral assembly. SH, G and F form the viral coat. The G protein is a surface protein; it functions as the attachment protein, the protein which attaches the virus to target cells. The F protein is another important surface protein. RSV’s name comes from the fact that F proteins on the surface of the virus cause the cell membranes on nearby cells to merge, forming syncytia.
Syncytia are aggregates of cells that can form when cells are infected with certain types of viruses, notably HIV and paramyxoviruses such as RSV. During infection, viral fusion proteins used by the virus to enter the cell are transported to the cell surface where they can cause the host cell membrane to fuse with neighboring cells. This presumably works to the virus’s advantage, as aggregates of target cells provide more hosts for the virus to infect and multiply. F proteins also mediate viral fusion, allowing entry of the virus into the cell cytoplasm and also allowing the formation of syncytia. Antibodies directed at the F protein are neutralizing. M2 is the second matrix protein required for viral transcription; it encodes M2-1 (elongation factor) and M2-2 (transcription regulation), while L encodes the RNA polymerase. The phosphoprotein P is a cofactor for L. The genome is transcribed sequentially from NS1 to L with reduction in expression levels along its length.
Treatment of RSV is limited to supportive care, including oxygen therapy. Studies of nebulized hypertonic saline (HS) have shown that the “use of nebulized 3% HS is a safe, inexpensive, and effective treatment for infants hospitalized with moderately severe viral bronchiolitis” where “RSV accounts for the majority of viral bronchiolitis cases. ” Supportive care includes fluids and oxygen until the illness runs its course. Increased airflow, humidified and delivered via nasal cannula, may be supplied in order to reduce the effort required for respiration.
Key Points
- RSV is a single stranded virus, the genome of which encodes 11 proteins which play different roles in RSV infection.
- RSV can induce syncytia (aggregates of host cells), providing further fertile ground for RSV to propagate.
- There is no direct treatment of RSV except to mitigate the symptoms, giving the patient’s body time to fight off the infection.
Key Terms
- cannula : A hose or tube that connects directly to an oxygen (O2) bottle/source from the user’s nose, commonly used by aircraft pilots or others needing a direct oxygen breathing apparatus.
- prophylactic : A medicine that preserves or defends against disease; a preventive.
- hypertonic : Having a greater osmotic pressure than another.


4.3: Dating Planetary Surfaces
Learning Objectives
By the end of this section, you will be able to:
- Explain how astronomers can tell whether a planetary surface is geologically young or old
- Describe different methods for dating planets
How do we know the age of the surfaces we see on planets and moons? If a world has a surface (as opposed to being mostly gas and liquid), astronomers have developed some techniques for estimating how long ago that surface solidified. Note that the age of these surfaces is not necessarily the age of the planet as a whole. On geologically active objects (including Earth), vast outpourings of molten rock or the erosive effects of water and ice, which we call planet weathering, have erased evidence of earlier epochs and present us with only a relatively young surface for investigation.
Counting the Craters
One way to estimate the age of a surface is by counting the number of impact craters. This technique works because the rate at which impacts have occurred in the solar system has been roughly constant for several billion years. Thus, in the absence of forces to eliminate craters, the number of craters is simply proportional to the length of time the surface has been exposed. This technique has been applied successfully to many solid planets and moons (Figure \(\PageIndex{1}\)).
Bear in mind that crater counts can tell us only the time since the surface experienced a major change that could modify or erase preexisting craters. Estimating ages from crater counts is a little like walking along a sidewalk in a snowstorm after the snow has been falling steadily for a day or more. You may notice that in front of one house the snow is deep, while next door the sidewalk may be almost clear.  More likely, you conclude that Jones has recently swept the walk clean and Smith has not. Similarly, the numbers of craters indicate how long it has been since a planetary surface was last “swept clean” by ongoing lava flows or by molten materials ejected when a large impact happened nearby.
Still, astronomers can use the numbers of craters on different parts of the same world to provide important clues about how regions on that world evolved. On a given planet or moon, the more heavily cratered terrain will generally be older (that is, more time will have elapsed there since something swept the region clean).
Radioactive Rocks
Another way to trace the history of a solid world is to measure the age of individual rocks. After samples were brought back from the Moon by Apollo astronauts, the techniques that had been developed to date rocks on Earth were applied to rock samples from the Moon to establish a geological chronology for the Moon. Furthermore, a few samples of material from the Moon, Mars, and the large asteroid Vesta have fallen to Earth as meteorites and can be examined directly (see the chapter on Cosmic Samples and the Origin of the Solar System).
Scientists measure the age of rocks using the properties of natural radioactivity. Around the beginning of the twentieth century, physicists began to understand that some atomic nuclei are not stable but can split apart (decay) spontaneously into smaller nuclei. The process of radioactive decay involves the emission of particles such as electrons, or of radiation in the form of gamma rays (see the chapter on Radiation and Spectra).
For any one radioactive nucleus, it is not possible to predict when the decay process will happen. Such decay is random in nature, like the throw of dice: as gamblers have found all too often, it is impossible to say just when the dice will come up 7 or 11. But, for a very large number of dice tosses, we can calculate the odds that 7 or 11 will come up. Similarly, if we have a very large number of radioactive atoms of one type (say, uranium), there is a specific time period, called its half-life, during which the chances are fifty-fifty that decay will occur for any of the nuclei.
A particular nucleus may last a shorter or longer time than its half-life, but in a large sample, almost exactly half of the nuclei will have decayed after a time equal to one half-life. Half of the remaining nuclei will have decayed after two half-lives pass, leaving only one half of a half—or one quarter—of the original sample (Figure \(\PageIndex{2}\)).
If you had 1 gram of pure radioactive nuclei with a half-life of 100 years, then after 100 years you would have
1/2 gram; after 200 years, 1/4 gram; after 300 years, only 1/8 gram; and so forth. However, the material does not disappear. Instead, the radioactive atoms are replaced with their decay products. Sometimes the radioactive atoms are called parents and the decay products are called daughter elements.
In this way, radioactive elements with half-lives we have determined can provide accurate nuclear clocks. By comparing how much of a radioactive parent element is left in a rock to how much of its daughter products have accumulated, we can learn how long the decay process has been going on and hence how long ago the rock formed. Table \(\PageIndex{1}\) summarizes the decay reactions used most often to date lunar and terrestrial rocks.
| Table \(\PageIndex{1}\): Radioactive Decay Reaction Used to Date Rocks 1 | ||
|---|---|---|
| Parent | Daughter | Half-Life (billions of years) |
| Samarium-147 | Neodymium-143 | 106 |
| Rubidium-87 | Strontium-87 | 48.8 |
| Thorium-232 | Lead-208 | 14.0 |
| Uranium-238 | Lead-206 | 4.47 |
| Potassium-40 | Argon-40 | 1.31 |
When astronauts first flew to the Moon, one of their most important tasks was to bring back lunar rocks for radioactive age-dating. Until then, astronomers and geologists had no reliable way to measure the age of the lunar surface. Counting craters had let us calculate relative ages (for example, the heavily cratered lunar highlands were older than the dark lava plains), but scientists could not measure the actual age in years. Some thought that the ages were as young as those of Earth’s surface, which has been resurfaced by many geological events. For the Moon’s surface to be so young would imply active geology on our satellite. Only in 1969, when the first Apollo samples were dated, did we learn that the Moon is an ancient, geologically dead world. Using such dating techniques, we have been able to determine the ages of both Earth and the Moon: each was formed about 4.5 billion years ago (although, as we shall see, Earth probably formed earlier).
We should also note that the decay of radioactive nuclei generally releases energy in the form of heat. Although the energy from a single nucleus is not very large (in human terms), the enormous numbers of radioactive nuclei in a planet or moon (especially early in its existence) can be a significant source of internal energy for that world. Geologists estimate that about half of Earth’s current internal heat budget comes from the decay of radioactive isotopes in its interior.
Key Concepts and Summary
The ages of the surfaces of objects in the solar system can be estimated by counting craters: on a given world, a more heavily cratered region will generally be older than one that is less cratered. We can also use samples of rocks with radioactive elements in them to obtain the time since the layer in which the rock formed last solidified. The half-life of a radioactive element is the time it takes for half the sample to decay; we determine how many half-lives have passed by how much of a sample remains the radioactive element and how much has become the decay product. In this way, we have estimated the age of the Moon and Earth to be roughly 4.5 billion years.
Footnotes
1 The number after each element is its atomic weight, equal to the number of protons plus neutrons in its nucleus. This specifies the isotope of the element; different isotopes of the same element differ in the number of neutrons.
Glossary
- half-life
- time required for half of the radioactive atoms in a sample to disintegrate
- radioactivity
- process by which certain kinds of atomic nuclei decay naturally, with the spontaneous emission of subatomic particles and gamma rays
Contributors and Attributions
-
Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


6.4: Evaluating Site Characteristics
- Create a summary of the essential site characteristics to be evaluated
You have decided to open a new business, or a new location for your existing business. Several sites are available. How do you choose just the right one? We have talked about how to get the needed data, so now let’s summarize the really important aspects of location. You have, I am sure, heard the old saying, that the success of a business is all about “location, location, location,” and that is absolutely true!
Learn More
- “ Choosing a Retail Store Location ” by Shari Waters on The Balance: Small Business
- “ The Retail Site Selection Process ” by Bill R. Shelton, CEcD HLM on Buxton
Entrepreneur Magazine has an excellent list of things to look at before picking a retail location .
Some of the items in this list can be part of a good review of the census or a GIS review. Some of these need to be done in person, walking the area! Let’s take a look:
- Your style of operation. If you are going to start an upscale steak house in a neighborhood filled with dollar stores and fast food restaurants, you may want to rethink your choice. Your style doesn’t match the area! There may be reasons that no one has started a fancy restaurant there yet!
- Demographics. We have discussed this one in detail, but let’s take another look. Who are your customers, how important is it to be close to your business and is the economic base of the location stable? If you plan to start a business in an area where there is one primary industry, it may be difficult to operate if that industry has a downturn.
- Foot traffic. Can people get to your door? How many people pass by on foot each day? This one will vary based on your type of business, but it is definitely important to consider!
- Accessibility and parking. The most awesome business may struggle if there is not enough parking or your customers can’t access it easily. A daycare center on the second floor of a building with no elevator, or a fitness center for senior citizens without handicapped accessibility might be examples of this problem. Make sure your target market can easily get to you!
- Competition. So here is an issue that may be good or bad. If there are several restaurants in a particular area that are always busy and overflowing, it may be good to add another! Ah, but if there are already several barbershops that are not busy, it may not be good to hang out your shingle in that neighborhood. Who is the competition, how busy are they, and is there room for another similar business?
- Proximity to other business and services. May also be good or bad. As in number 5, competition may enhance your business or damage it. It may also be good to open a complimentary business to those already in the area. For example a drop-in child care center next to a fitness center or spa might be very popular and needed!
- The image and history of the site. A good history of successful businesses in a spot might be amazing! But what if the space you have chosen has had 5 different restaurants in as many years? Was it the location? The food? Poor management? It could have been ANY of these things, and you should delve deep to find out!
- Ordinances. Sometimes to get a space up to par is a challenge in a particular community or neighborhood. If you are interested in opening a coffee shop and would like to have outdoor seating and serve wine and beer, check with the town or city to insure that these things are options, and what the fees may be. A great location, isn’t great if you can’t do the things you need due to regulations, licensing and permit issues!! I recently had a conversation with a guy who bought a building to put small rental rooms into, but he found out it wasn’t permissible in this location! He now needs to find a buyer for a building that has limited uses. A few phone calls may have averted this issue!
- The condition of the building. It may be difficult to put a high tech computer store in a building with old wiring, or a laundromat in a building without sufficient water hook ups. Make sure that the building you are looking at will fit your needs from an infrastructure standpoint. Have it professionally inspected, and find out how much renovations may cost to make it work.
- All the other costs. Utilities, insurance, janitorial service, refuse removal and parking fees can all add up! That building with a low lease cost may cost more than you could imagine once you add all of those costs in. Find out what is included in the lease, and research the other costs before you jump in!
So, there is a lot to consider when you think about picking a retail location!
Contributors and Attributions
- Evaluating Site Characteristics. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution