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\(\def\R{{\mathbb R}} \def\C{{\mathbb C}} \def\Q{{\mathbb Q}} \def\Z{{\mathbb Z}} \def\N{{\mathbb N}} \def\a{&} \newcommand{\adj}{\mathop{\textrm{adj}}} \newcommand{\proj}{\mathop{\textrm{proj}}} \newcommand{\rowint}[2]{R_{#1} \leftrightarrow R_{#2}} \newcommand{\rowmul}[2]{R_{#1}\gets {#2}R_{#1}} \newcommand{\rowadd}[3]{R_{#1}\gets R_{#1}+#2R_{#3}} \newcommand{\rowsub}[3]{R_{#1}\gets R_{#1}-#2R_{#3}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Chapter9Textbook exercise solutions

The number before the solution corresponds to the page and problem number in the textbook.

Section9.1Page 50 and following

50-20
Solution
50-21
Solution
51-22
Solution
51-23
Solution
51-24
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52-32
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52-33
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52-34
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52-35
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52-36
Solution
54-48
Solution
55-49
Solution
55-50
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55-51
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55-52
Solution

Section9.2Page 97

97-2
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97-4
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98-9
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98-10
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99-11
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101-27
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101-35
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102-36
Solution
102-37
Solution
102-38
Solution
102-40
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102-41
Solution
102-42
Solution
102-44
Solution
104-54
Solution
104-55
Solution
104-56
Solution
104-57
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104-58
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104-59
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105-60
Solution

Section9.3Page 121

121-1
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122-2
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122-3
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122-4
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122-5
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122-6
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122-7
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123-8
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123-9
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123-10
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123-11
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123-12
Solution

Section9.4Page 136

136-1
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136-2
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137-3
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137-4
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137-5
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139-18
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139-19
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Section9.5Page 147

147-1
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147-2
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Section9.6Page 161

161-1
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161-2
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161-3
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161-4
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161-5
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161-6
Solution
161-7
Solution
161-8
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161-9
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161-10
Solution
161-11
Hint Solution
161-extra

<<SVG image is unavailable, or your browser cannot render it>>

Let \(B=(2,1,4,3)\) and \(C=(1,2,3,4)\) be points in \(\R^4\text{.}\) and let \(L\) be the line joining \(B\) and \(C\text{.}\) The points \(A\) and \(D\) are also on \(L\) with the distances from \(A\) to \(B\text{,}\) from \(B\) to \(C\) and from \(C\) to \(D\) all equal to \(d\text{.}\)

  1. What is the value of \(d\text{?}\)
  2. Find \(A\) and \(D\)

Hint Solution

Section9.7Page 172 and following

172-8
Solution
172-9
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172-10
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172-11
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172-12
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172-13
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172-14
Solution

Section9.8Page 186 and following

186-2
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186-3
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186-4
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186-5
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186-6
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186-10
Solution
186-extra-1

Consider vectors in \(\R^4\text{.}\)

  1. Find the distance from the point \(\mathbf{p}=(1,0,1,0)\) to the hyperplane \(x_1+2x_2+x_3-x_4=7\text{.}\)

  2. Find the point \(\mathbf{y}\) in the hyperplane that is closest to \(\mathbf{p}\text{.}\)

Solution
186-extra-2

Let \(\mathbf{p}=(1,1,1,1)\) and let the line \(L\) be the points of the form \((1,1,2,1)+t(1,-1,-1,1)\) where \(t\) is any real number. Find the point on \(L\) closest to \(\mathbf{p}\text{.}\)

Solution 1 Solution 2

Section9.9Page 303 and following

1

Suppose \(A\mathbf x=\lambda\mathbf x\) and \(A\mathbf y=\mu\mathbf y\)

  1. Show that \(A(r\mathbf x)=\lambda r \mathbf x\) for any real number \(r\) (and so \(r\mathbf x\) is also an eigenvector).
  2. Show that \(A^2\mathbf x=\lambda^2\mathbf x\) (and so \(\mathbf x\) is also an eigenvector of \(A^2\)).
  3. Show that \(A(r\mathbf x + s\mathbf y)=r\lambda \mathbf x+s\mu\mathbf y\)

Solution
304-6
Solution
304-7
Solution
305-8
Solution
305-9
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305-10
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305-11
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305-12
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306-13
Solution

Section9.10Extra problems

1

Let \(P=(-1,1,1)\text{,}\) \(Q=(1,2,3)\) and \(R=(2,1,0)\) be points in \(\R^3\text{.}\)

  1. Find the equation of the plane in \(\R^3\) containing \(P\text{,}\) \(Q\) and \(R\text{.}\)

  2. Find the equation of line \(L\) in \(\R^3\) passing through \(P\) and \(Q\text{.}\)

  3. Find the area \(A\) of the triangle determined by \(P\text{,}\) \(Q\) and \(R\text{.}\)

  4. Find the point of intersection of \(L\) and the \(xy\)-plane.

Solution
2

Let \begin{equation*} A=\begin{bmatrix} 1\amp 1\amp 0\amp 1\\ 0\amp 1\amp 0\amp 1\\ 1\amp 1\amp 1\amp 0\\ 0\amp 0\amp 0\amp 1 \end{bmatrix} \end{equation*}

  1. Find the determinant of \(A\text{.}\)

  2. Find \(A^{-1}\text{.}\)

  3. Find all solutions \(\mathbf x\) to \[ A\mathbf x=\begin{bmatrix} 1\\2\\2\\1 \end{bmatrix} \]

  4. Find the adjoint of \(A\text{.}\)

Solution
3

Let \begin{equation*} A= \begin{bmatrix} 1\amp0\amp0\\ 1\amp1\amp1\\ 0\amp0\amp-1 \end{bmatrix} \end{equation*} Evaluate the following matrices:

  1. \(A^T\)

  2. \(A^{-1}\)

  3. \((A^{-1})^T\)

  4. \((A^T)^{-1}\)

  5. \((A^{-1})^2\)

  6. \((A^2)^{-1}\)

Solution
4

  1. Find a unit vector in the direction of \(\mathbf w=(4,-3,2,1)\text{.}\)

  2. Find all values \(t\) so that vectors \((2,5,-3,6)\) and \((4,t,7,1)\) are orthogonal.

  3. Calculate \(\proj_{\mathbf v} \mathbf u\text{,}\) the projection of \(\mathbf u\) along \(\mathbf v\text{,}\) where \(\mathbf u=(2,-2,3)\) and \(\mathbf v=(2,1,3)\text{.}\)

Solution
5

Let \(\mathbf{u}=(1,1,1,-1)\text{,}\) \(\mathbf{v}=(1,1,-1,-1)\) and \(\mathbf{w}=(1,1,1,1)\) be three vectors in 4-space.

    1. What is the length \(\|\mathbf{u}\|\)

    2. What is the length \(\|\mathbf{v}\|\)

    3. What is the length \(\|\mathbf{w}\|\)

    1. What is the angle between \(\mathbf{u}\) and \(\mathbf{v}\)

    2. What is the angle between \(\mathbf{u}\) and \(\mathbf{w}\)

    3. What is the angle between \(\mathbf{v}\) and \(\mathbf{w}\)

  1. Find a vector in 4-space that is orthogonal to \(\mathbf{u}\text{,}\) \(\mathbf{v}\) and \(\mathbf{w}\text{.}\)

Solution
6

Consider the following system of linear equations: \[ \begin{array}{rrrrrr} x \amp +2y \amp +z \amp +w \amp =\amp 2\\ 2x \amp -y \amp +z \amp \amp =\amp 1\\ -3x \amp -y \amp -2z \amp -w \amp =\amp -3\\ x \amp +7y \amp +2z \amp +3w \amp =\amp 5 \end{array} \]

  1. Give the augmented matrix of the system.

  2. Put the augmented matrix in reduced row echelon form.

  3. Give all of the solutions to the system.

Solution
7

Let \begin{equation*} A= \begin{bmatrix} 1 \amp 2 \amp 1 \\ 0 \amp 1 \amp 1 \\ 1 \amp 1 \amp 1 \end{bmatrix} \hbox{ and } B= \begin{bmatrix} 1\amp 2\\2\amp 1\\1\amp 1 \end{bmatrix} \end{equation*} Find the matrix \(X\) so that \(AX=B\text{.}\)

Solution
8

Let \[ A= \begin{bmatrix} 2\amp 0\amp 4\\ -1\amp 0\amp -2\\ 1\amp -4\amp -2 \end{bmatrix} \]

  1. Find all the eigenvalues of \(A\text{.}\)

  2. Find an eigenvector for the each positive eigenvalue.

  3. Show that your eigenvector satisfies \(A\mathbf x =\lambda\mathbf x\text{.}\)

Solution
9

Let \(\Pi\) be the plane in \(\R^3\) with equation \(x+2y-z=6\) and let \(P=(1,-1,-1)\text{.}\)

  1. Find the distance from the point \(P\) to the plane \(\Pi\text{.}\)

  2. Find the equation of the line \(L\) through \(P\) and perpendicular to \(\Pi\text{.}\)

  3. Find the point \(Q\text{,}\) the intersection of the line \(L\) and plane \(\Pi\text{.}\)

  4. Compute the distance from \(P\) to \(Q\text{.}\)

Solution
10

In each case, either prove that the statement is true or give an example to show that it is false:

  1. The matrices \(A\) and \(-A\) have the same reduced row echelon form.

  2. The matrices \(A\) and \(-A\) have the same determinant.

  3. The matrices \(A\) and \(-A\) have the same eigenvalues.

  4. The matrices \(A\) and \(-A\) have the same rank.

Solution