# Chapter9Textbook exercise solutions

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

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# Section9.6Page 161

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##### 161-extra

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 $D$

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# Section9.8Page 186 and following

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##### 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 $\mathbf{p}\text{.}$

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##### 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 $\mathbf{p}\text{.}$

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# 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 $A(r\mathbf x + s\mathbf y)=r\lambda \mathbf x+s\mu\mathbf y$

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# 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.

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##### 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 $A\text{.}$

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##### 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. $(A^2)^{-1}$

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##### 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 $\mathbf v=(2,1,3)\text{.}$

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##### 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 $\mathbf{w}\text{.}$

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##### 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.

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##### 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 $AX=B\text{.}$

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##### 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 $A\mathbf x =\lambda\mathbf x\text{.}$

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##### 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 $Q\text{.}$

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##### 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.

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