MATH 2160 — Question Compendium

Every question from the 23 completed MyLab assignments (Lay, Linear Algebra and Its Applications, 6e), transcribed to LaTeX with the recorded answer. Linked from the chapter study guides.

343 questions across 22 sections.

§1.1 Systems of Linear Equations

1.1.1

Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure. $x_{1} + 5 x_{2} = 8 2 x_{1} + 7 x_{2} = 7$

1.1.2

Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure. $7 x_{1} + 14 x_{2} = 28 3 x_{1} + 7 x_{2} = 9$

1.1.3

Find the point $(x_{1}, x_{2})$ that lies on the line $x_{1} + 3 x_{2} = 10$ and on the line $x_{1} - x_{2} = 2$. See the figure. $x_{1} x_{2}$ A graph has a horizontal axis labeled $x_{1}$ and a vertical axis labeled $x_{2}$. A line on the graph falls from left to right and another line rises from left to right. The lines intersect in the first quadrant. The lines are not labeled and the axes do not show any scale

1.1.6

Consider the accompanying matrix as the augmented matrix of a linear system. State in words the

1.1.7

The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system.$\begin{bmatrix} 1 & 8 & 4 & -3 \\ 0 & 0 & 0 & -3 \\ 0 & 0 & 1 & 2 \\ 0 & 1 & -1 & 2 \end{bmatrix}$

1.1.10

The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system.$\begin{bmatrix} 1 & -2 & 0 & 0 & -4 \\ 0 & 1 & -1 & 0 & -5 \\ 0 & 0 & 1 & -3 & 2 \\ 0 & 0 & 0 & 1 & 4 \end{bmatrix}$

1.1.13

Solve the system. $x_{1} - 6 x_{3} = 17 4 x_{1} + 2 x_{2} - 9 x_{3} = 44 x_{2} + 5 x_{3} = -7$

1.1.22

Do the three planes $x_{1} + 3 x_{2} + x_{3} = 5, x_{2} - x_{3} = 1$, and $x_{1} + 4 x_{2} = 4$ have at least one common point of intersection? Explain

1.1.27

Determine whether the statement below is true or false. Justify the answer. Every elementary row operation is reversible

1.1.28

Determine whether the statement below is true or false. Justify the answer. Elementary row operations on an augmented matrix never change the solution set of the associated linear system

1.1.29

Determine whether the statement below is true or false. Justify the answer. $A_{5} \times 6$ matrix has six rows

1.1.30

Determine whether the statement below is true or false. Justify the answer. Two matrices are row equivalent if they have the same number of rows

1.1.31

Determine whether the statement below is true or false. Justify the answer. The solution set of a linear system involving variables $x_{1}$,…, $x_{n}$ is a list of numbers $(s_{1}$,…, $s_{n})$ that makes each equation in the system a true statement when the values $s_{1}$,…, $s_{n}$ are substituted for $x_{1}$,…, $x_{n}$, respectively

1.1.39

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse operation that transforms the second matrix into the first.$\begin{bmatrix} -1 & 6 & -1 \\ 2 & -2 & 1 \\ -4 & 6 & -3 \end{bmatrix}$,$\begin{bmatrix} -4 & 6 & -3 \\ 2 & -2 & 1 \\ -1 & 6 & -1 \end{bmatrix}$

1.1.41

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.$\begin{bmatrix} 1 & -4 & 2 & 0 \\ 2 & -2 & 3 & -6 \\ 0 & 4 & -3 & 5 \end{bmatrix}$,$\begin{bmatrix} 1 & -4 & 2 & 0 \\ 0 & 6 & -1 & -6 \\ 0 & 4 & -3 & 5 \end{bmatrix}$

§1.2 Row Reduction and Echelon Forms

1.2.2

Determine which matrices are in reduced echelon form and which others are only in echelon form. a.$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$b.$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$c.$\begin{bmatrix} 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

1.2.3

Row reduce the matrix to reduced echelon form. Identify the pivot positions in the final matrix and in the original matrix, and list the pivot columns.$\begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 5 & 6 & 7 \\ 7 & 8 & 9 & 10 \end{bmatrix}$

1.2.4

Row reduce the matrix to reduced echelon form. Identify the pivot positions in the final matrix and in the original matrix, and list the pivot columns.$\begin{bmatrix} 1 & 2 & 4 & 1 \\ 2 & 4 & 5 & -1 \\ 4 & 5 & 4 & -2 \end{bmatrix}$

1.2.8

Find the general solution of the system whose augmented matrix is given below.$\begin{bmatrix} 1 & 3 & 0 & 10 \\ 2 & 5 & 0 & 15 \end{bmatrix}$

1.2.9

Find the general solution of the system whose augmented matrix is given below.$\begin{bmatrix} 0 & 1 & -2 & 3 \\ 1 & -3 & 1 & -3 \end{bmatrix}$

1.2.11

Find the general solution of the system whose augmented matrix is given below.$\begin{bmatrix} 2 & -3 & 5 & 0 \\ 8 & -12 & 20 & 0 \\ 6 & -9 & 15 & 0 \end{bmatrix}$

1.2.13

Find the general solution of the system whose augmented matrix is given below.$\begin{bmatrix} 1 & -6 & 0 & -1 & 0 & -8 \\ 0 & 1 & 0 & 0 & -5 & 4 \\ 0 & 0 & 0 & 1 & 8 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

1.2.14

Find the general solution of the system whose augmented matrix is given below.$\begin{bmatrix} 1 & 0 & -4 & 0 & -7 & 2 \\ 0 & 1 & 5 & -1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

1.2.25

Determine whether the statement below is true or false. Justify the answer. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations

1.2.26

Determine whether the statement below is true or false. Justify the answer. The echelon form of a matrix is unique

1.2.34

Determine whether the statement below is true or false. Justify the answer. A general solution of a system is an explicit description of all solutions of the system

1.2.35

Suppose $a_{5} \times 7$ coefficient matrix for a system has five pivot columns. Is the system consistent? Why or why not?

1.2.36

Suppose a system of linear equations has $a_{3} \times 5$ augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not?

1.1

Solve the system of equations. $2 x_{1} + x_{2} = 0 x_{1} - 3 x_{2} + x_{3} = 0 3 x_{1} + x_{2} - x_{3} = 0$

§1.3 Vector Equations

1.3.1

Compute$\mathbf{u}$+$\mathbf{v}$and$\mathbf{u} - 4 \mathbf{v}$.$\mathbf{u}$=$\begin{bmatrix} -6 \\ 4 \end{bmatrix}$,$\mathbf{v}$=$\begin{bmatrix} -3 \\ 3 \end{bmatrix}$

1.3.5

Write a system of equations that is equivalent to the given vector equation. $x_{1} \begin{bmatrix} 5 \\ -5 \\ 9 \end{bmatrix} + x_{2} \begin{bmatrix} 4 \\ 0 \\ -9 \end{bmatrix}$=$\begin{bmatrix} 4 \\ -2 \\ 6 \end{bmatrix}$

1.3.6

Write a system of equations that is equivalent to the given vector equation. $x_{1} \begin{bmatrix} 5 \\ -4 \end{bmatrix} + x_{2} \begin{bmatrix} 9 \\ 5 \end{bmatrix} + x_{3} \begin{bmatrix} -4 \\ 1 \end{bmatrix}$=$\begin{bmatrix} 0 \\ 0 \end{bmatrix}$

1.3.7

Use the accompanying figure to write each vector listed as a linear combination of u and v. Vectors a, w, x, and z

1.3.9

Write a vector equation that is equivalent to the given system of equations. $x_{2} + 4 x_{3} = 0 4 x_{1} + 7 x_{2} - x_{3} = 0$ negative $x_{1} + 3 x_{2} - 6 x_{3} = 0$

1.3.11

Determine if b is a linear combination of $a_{1}, a_{2}$, and $a_{3}$. $a_{1} = \begin{bmatrix} 1 \\ -3 \\ 0 \end{bmatrix}$, $a_{2} = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$, $a_{3} = \begin{bmatrix} 5 \\ -4 \\ 22 \end{bmatrix}$, $b = \begin{bmatrix} 4 \\ -3 \\ 18 \end{bmatrix}$

1.3.13

Determine if b is a linear combination of the vectors formed from the columns of the matrix A. $A = \begin{bmatrix} 1 & -6 & 5 \\ 0 & 5 & 6 \\ -3 & 18 & -15 \end{bmatrix}$, $b = \begin{bmatrix} 4 \\ -6 \\ -4 \end{bmatrix}$

1.3.14

Determine if b is a linear combination of the vectors formed from the columns of the matrix A. $A = \begin{bmatrix} 1 & -6 & -4 \\ 0 & 4 & 3 \\ 3 & -18 & 11 \end{bmatrix}$,$\mathbf{b}$=$\begin{bmatrix} 12 \\ -5 \\ 7 \end{bmatrix}$

1.3.17

Let $a_{1} = \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}$, $a_{2} = \begin{bmatrix} -7 \\ -31 \\ 3 \end{bmatrix}$, and $b = \begin{bmatrix} 5 \\ 13 \\ h \end{bmatrix}$. For what value(s) of h is b in the plane spanned by $a_{1}$ and $a_{2}$ ?

1.3.23

Determine whether the statement below is true or false. Justify the answer. Another notation for the vector$\begin{bmatrix} -4 \\ 3 \end{bmatrix}$is$\begin{bmatrix} -4 & 3 \end{bmatrix}$

1.3.27

Determine whether the statement below is true or false. Justify the answer. An example of a linear combination of vectors$\mathbf{v}_{1}$and$\mathbf{v}_{2}$is the vector one half$\mathbf{v}_{1}$

1.3.29

Determine whether the statement below is true or false. Justify the answer. The solution set of the linear system whose augmented matrix is$\begin{bmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} & \mathbf{b} \end{bmatrix}$is the same as the solution set of the equation $x_{1} \mathbf{a}_{1} + x_{2} \mathbf{a}_{2} + x_{3} \mathbf{a}_{3}$=$\mathbf{b}$

1.3.31

Determine whether the statement below is true or false. Justify the answer. The set Span $\{ \mathbf{u}$,$\mathbf{v} \}$ is always visualized as a plane through the origin

1.3.32

Determine whether the statement below is true or false. Justify the answer. Asking whether the linear system corresponding to an augmented matrix$\begin{bmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} & \mathbf{b} \end{bmatrix}$has a solution amounts to asking whether$\mathbf{b}$is in Span $\{ \mathbf{a}_{1}$,$\mathbf{a}_{2}$,$\mathbf{a}_{3} \}$

1.3.33

Let $A = \begin{bmatrix} 1 & 0 & -3 \\ 0 & 2 & -6 \\ -2 & 6 & 4 \end{bmatrix}$and$\mathbf{b}$=$\begin{bmatrix} 3 \\ 4 \\ -26 \end{bmatrix}$. Denote the columns of A by$\mathbf{a}_{1}$,$\mathbf{a}_{2}$,$\mathbf{a}_{3}$, and let $W =$ Span $\{ \mathbf{a}_{1}$,$\mathbf{a}_{2}$,$\mathbf{a}_{3} \}$. a. Is$\mathbf{b}$in $\{ \mathbf{a}_{1}$,$\mathbf{a}_{2}$,$\mathbf{a}_{3} \}$ ? How many vectors are in $\{ \mathbf{a}_{1}$,$\mathbf{a}_{2}$,$\mathbf{a}_{3} \}$ ? b. Is$\mathbf{b}$in W? How many vectors are in W? c. Show that$\mathbf{a}_{2}$is in W. [ Hint: Row operations are unnecessary.]

1.3.37

Let$\mathbf{v}_{1}$,…., $\mathbf{v}_{k}$ be the points in $\mathbb{R}^{3}$ and suppose that for $j = 1$,…, k an object with mass $m_{j}$ is located at point $\mathbf{v}_{j}$. Physicists call such objects point masses. The total mass of the system of point masses is $m = m_{1} + \cdots + m_{k}$. The center of gravity (or center of mass) of the system is$\mathbf{v}$overbar $= \frac{1}{m} m_{1} \mathbf{v}_{1} + \cdots + m_{k} \mathbf{v}_{k}$. Point Mass$\mathbf{v}_{1} = (4, -4, 3) 8 g \mathbf{v}_{2} = (6, 2, -1) 20 g \mathbf{v}_{3}$= (-4, -3, -1) 8 g$\mathbf{v}_{4}$= (-9, 6, 6) 4 g Compute the center of gravity of the system consisting of the point masses above

1.3.42

Use the vector$\mathbf{u} = (u_{1}$,…, $u_{n})$ to verify the following algebraic properties of $\mathbb{R}^{n}$. a.$\mathbf{u}$+ (-$\mathbf{u}$) = (-$\mathbf{u}$) +$\mathbf{u}$=$\mathbf{0}$b. c(d$\mathbf{u}$) = (cd)$\mathbf{u}$for all scalars c and d See Sample Answer

1.3

Find the indicated vector. Let $u = \begin{bmatrix} 6 \\ -8 \end{bmatrix}$, $v = \begin{bmatrix} 6 \\ 9 \end{bmatrix}$. Find $u - v$

1.3.28

Determine whether the statement below is true or false. Justify the answer. The weights $c_{1}$,…, $c_{p}$ in a linear combination $c_{1} \mathbf{v}_{1} + \cdots + c_{p} \mathbf{v}_{p}$ cannot all be zero

§1.4 The Matrix Equation Ax = b

1.4.1

Compute the product using (a) the definition where A x is the linear combination of the columns of A using the corresponding entries in x as weights, and (b) the row-vector rule for computing A x. If a product is undefined, explain why.$\begin{bmatrix} -1 & 2 \\ 2 & 9 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 2 \\ -10 \\ 6 \end{bmatrix}$

1.4.4

Compute the product using the methods below. If a product is undefined, explain why. a. The definition where A x is the linear combination of the columns of A using the corresponding entries in x as weights. b. The row-vector rule for computing A x.$\begin{bmatrix} 7 & 5 & -6 \\ 3 & 1 & 3 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$

1.4.6

Use the definition of A x to write the matrix equation as a vector equation.$\begin{bmatrix} 5 & -6 \\ -9 & 6 \\ -8 & 0 \\ -8 & -5 \end{bmatrix} \begin{bmatrix} -3 \\ 3 \end{bmatrix}$=$\begin{bmatrix} -33 \\ 45 \\ 24 \\ 9 \end{bmatrix}$

1.4.8

Use the definition of A x to write the vector equation as a matrix equation. $z_{1} \begin{bmatrix} 3 \\ 3 \end{bmatrix} + z_{2} \begin{bmatrix} -1 \\ -4 \end{bmatrix} + z_{3} \begin{bmatrix} -1 \\ 4 \end{bmatrix} + z_{4} \begin{bmatrix} -2 \\ 1 \end{bmatrix}$=$\begin{bmatrix} -2 \\ 2 \end{bmatrix}$

1.4.9

Write the system first as a vector equation and then as a matrix equation. $3 x_{1} + x_{2} - 3 x_{3} = 5 7 x_{2} + 4 x_{3} = 0$

1.4.10

Write the system first as a vector equation and then as a matrix equation. $3 x_{1} - x_{2} = 3 10 x_{1} + 5 x_{2} = 3 6 x_{1} - x_{2} = 1$

1.4.12

Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation $A x = b$. Then solve the system and write the solution as a vector. $A = \begin{bmatrix} 1 & 5 & -7 \\ 1 & 4 & 2 \\ -4 & -7 & 2 \end{bmatrix}$, $b = \begin{bmatrix} -2 \\ 13 \\ -5 \end{bmatrix}$

1.4.13

Let $u = \begin{bmatrix} -21 \\ 27 \\ 6 \end{bmatrix}$and $A = \begin{bmatrix} 4 & -6 \\ -3 & 7 \\ 1 & 1 \end{bmatrix}$. Is u in the plane in $\mathbb{R}^{3}$ spanned by the columns of A? Why or why not?

1.4.14

Let $u = \begin{bmatrix} 1 \\ -1 \\ 9 \end{bmatrix}$and $A = \begin{bmatrix} 5 & 8 & 12 \\ 1 & 2 & 2 \\ 2 & 1 & 7 \end{bmatrix}$. Is u in the subset of $\mathbb{R}^{3}$ spanned by the columns of A? Why or why not?

1.4.15

Let $A = \begin{bmatrix} 2 & -4 \\ -8 & 16 \end{bmatrix}$and $b = \begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix}$. Show that the equation $A x = b$ does not have a solution for some choices of b, and describe the set of all b for which $A x = b$ does have a solution

1.4.16

Let $A = \begin{bmatrix} 1 & -3 & -2 \\ -4 & 4 & 0 \\ 3 & -1 & 2 \end{bmatrix}$and $b = \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}$. Show that the equation $A x = b$ does not have a solution for all possible b, and describe the set of all b for which $A x = b$ does have a solution

1.4.17

How many rows of A contain a pivot position? Does the equation $A x = b$ have a solution for each b in $\mathbb{R}^{4}$ ? $A = \begin{bmatrix} 2 & 4 & 1 & 2 \\ -2 & -2 & -2 & 2 \\ 0 & -4 & 2 & -8 \\ 2 & 0 & 3 & -3 \end{bmatrix}$

1.4.21

Let$\mathbf{v}_{1}$=$\begin{bmatrix} 0 \\ 0 \\ -1 \\ 1 \end{bmatrix}$,$\mathbf{v}_{2}$=$\begin{bmatrix} 1 \\ -1 \\ 0 \\ 0 \end{bmatrix}$, and$\mathbf{v}_{3}$=$\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \end{bmatrix}$. Does $\{ \mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3} \}$ span $\mathbb{R}^{4}$ ? Why or why not?

1.4.22

Let$\mathbf{v}_{1}$=$\begin{bmatrix} 0 \\ 0 \\ -3 \end{bmatrix}$,$\mathbf{v}_{2}$=$\begin{bmatrix} 0 \\ -2 \\ -6 \end{bmatrix}$, and$\mathbf{v}_{3}$=$\begin{bmatrix} 6 \\ -3 \\ 9 \end{bmatrix}$. Does $\{ \mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3} \}$ span $\mathbb{R}^{3}$ ? Why or why not?

1.4.23

Determine whether the statement below is true or false. Justify the answer. The equation $A x = b$ is referred to as a vector equation

1.4.24

Determine whether the statement below is true or false. Justify the answer. Every matrix equation $A x = b$ corresponds to a vector equation with the same solution set

1.4.26

Determine whether the statement below is true or false. Justify the answer. A vector b is a linear combination of the columns of a matrix A if and only if the equation $A x = b$ has at least one solution

1.4.31

Determine whether the statement below is true or false. Justify the answer. If the columns of an $m \times n$ matrix A span $\mathbb{R}^{m}$, then the equation $A x = b$ is consistent for each b in $\mathbb{R}^{m}$

1.4.32

Determine whether the statement below is true or false. Justify the answer. The solution set of a linear system whose augmented matrix is$\begin{bmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} & \mathbf{b} \end{bmatrix}$is the same as the solution set of $A x = b$, if A=$\begin{bmatrix} \mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3} \end{bmatrix}$

1.4.42

Could a set of three vectors in $\mathbb{R}^{4}$ span all of $\mathbb{R}^{4}$ ? Explain. What

§1.5 Solution Sets of Linear Systems

1.5.1

Determine if the system has a nontrivial solution. Try to use as few row operations as possible. $6 x_{1} - 3 x_{2} + 15 x_{3} = 0 -6 x_{1} - 9 x_{2} - 6 x_{3} = 0 12 x_{1} + 6 x_{2} + 21 x_{3} = 0$

1.5.2

Determine if the system has a nontrivial solution. Try to use as few row operations as possible. $x_{1} - 4 x_{2} + 6 x_{3} = 0 - 2 x_{1} + 4 x_{2} - 3 x_{3} = 0 x_{1} +$ plus $10 x_{3} = 0$

1.5.3

Determine if the system has a nontrivial solution. Try to use as few row operations as possible. $-4 x_{1} + 6 x_{2} - 5 x_{3} = 0 -8 x_{1} + 9 x_{2} + 4 x_{3} = 0$

1.5.8

Describe all solutions of $A x = 0$ in parametric vector form, where A is row equivalent to the given matrix.$\begin{bmatrix} 1 & -4 & -6 & 4 \\ 0 & 1 & 2 & -7 \end{bmatrix}$

1.5.9

Describe all solutions of $A x = 0$ in parametric vector form, where A is row equivalent to the given matrix.$\begin{bmatrix} 4 & -16 & 20 \\ -1 & 4 & -5 \end{bmatrix}$

1.5.10

Describe all solutions of $A x = 0$ in parametric vector form, where A is row equivalent to the given matrix.$\begin{bmatrix} 1 & 4 & 0 & -2 \\ 4 & 16 & 0 & -8 \end{bmatrix}$

1.5.17

Suppose the solution set of a certain system of linear equations can be described as $x_{1} = 4 + 5 x_{3}, x_{2} = -5 - 7 x_{3}$, with $x_{3}$ free. Use vectors to describe this set as a line in $\mathbb{R}^{3}$

1.5.23

Find the parametric equation of the line through a parallel to b, using t as the parameter. $a = \begin{bmatrix} -5 \\ 2 \end{bmatrix}$, $b = \begin{bmatrix} -6 \\ 3 \end{bmatrix}$

1.5.27

Determine whether the statement below is true or false. Justify the answer. A homogeneous equation is always consistent

1.5.28

Determine whether the statement below is true or false. Justify the answer. If x is a nontrivial solution of $A x = 0$, then every entry in x is nonzero

1.5.31

Determine whether the statement below is true or false. Justify the answer. The homogenous equation $A x = 0$ has the trivial solution if and only if the equation has at least one free variable

1.5.32

Determine whether the statement below is true or false. Justify the answer. The equation $A x = b$ is homogeneous if the zero vector is a solution

1.5.34

Determine whether the statement below is true or false. Justify the answer. The effect of adding p to a vector is to move the vector in a direction parallel to p

1.5.41

A is $a_{3} \times 3$ matrix with three pivot positions. (a) Does the equation $A x = 0$ have a nontrivial solution? (b) Does the equation $A x = b$ have at least one solution for every possible b ?

1.5.42

Let A be $a_{3} \times 3$ matrix with two pivot positions. Use this information to answer parts (a) and (b) below

1.5.44

A is $a_{2} \times 5$ matrix with two pivot positions. (a) Does the equation $A x = 0$ have a nontrivial solution? (b) Does the equation $A x = b$ have at least one solution for every possible b ?

§1.7 Linear Independence

1.7.2

Determine if the vectors are linearly independent. Justify your answer.$\begin{bmatrix} 7 \\ 0 \\ 0 \end{bmatrix}$,$\begin{bmatrix} 9 \\ 3 \\ -12 \end{bmatrix}$,$\begin{bmatrix} 8 \\ 12 \\ -24 \end{bmatrix}$

1.7.4

Determine if the vectors are linearly independent.$\mathbf{v}_{1}$=$\begin{bmatrix} 1 \\ -2 \end{bmatrix}$,$\mathbf{v}_{2}$=$\begin{bmatrix} -2 \\ 4 \end{bmatrix}$

1.7.5

Determine if the columns of the matrix form a linearly independent set. Justify your answer.$\begin{bmatrix} 0 & -8 & 16 \\ 3 & 1 & -14 \\ -1 & 5 & -5 \\ 1 & -5 & -2 \end{bmatrix}$

1.7.6

Determine if the columns of the matrix form a linearly independent set. Justify your answer.$\begin{bmatrix} -2 & -1 & 0 \\ 0 & -1 & 4 \\ 1 & 1 & -8 \\ 2 & 1 & -16 \end{bmatrix}$

1.7.7

Determine if the columns of the matrix form a linearly independent set.$\begin{bmatrix} 1 & 3 & -3 & 1 \\ 2 & 7 & -4 & -2 \\ 2 & 8 & 1 & -12 \end{bmatrix}$

1.7.8

Determine if the columns of the matrix form a linearly independent set. Justify your answer.$\begin{bmatrix} 1 & -2 & 3 & 4 \\ -2 & 4 & -6 & 4 \\ 0 & 1 & -1 & 5 \end{bmatrix}$

1.7.15

Determine by inspection whether the vectors are linearly independent. Justify your answer.$\begin{bmatrix} 4 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 3 \\ 8 \end{bmatrix}$,$\begin{bmatrix} 1 \\ 3 \end{bmatrix}$,$\begin{bmatrix} -1 \\ 6 \end{bmatrix}$

1.7.16

Determine by inspection whether the vectors are linearly independent. Justify your answer.$\begin{bmatrix} 3 \\ -9 \\ 6 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -6 \\ 4 \end{bmatrix}$

1.7.17

Determine by inspection whether the vectors are linearly independent. Justify your answer.$\begin{bmatrix} 6 \\ 4 \\ -1 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$,$\begin{bmatrix} -3 \\ 3 \\ 4 \end{bmatrix}$

1.7.19

Determine by inspection whether the vectors are linearly independent. Justify your answer.$\begin{bmatrix} -6 \\ -12 \\ -3 \end{bmatrix}$,$\begin{bmatrix} 2 \\ 4 \\ -1 \end{bmatrix}$

1.7.21

Determine whether the statement below is true or false. Justify the answer. The columns of a matrix A are linearly independent if the equation $A x = 0$ has the trivial solution

1.7.22

Determine whether the statement below is true or false. Justify the answer. Two vectors are linearly dependent if and only if they lie on a line through the origin

1.7.23

Determine whether the statement below is true or false. Justify the answer. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S

1.7.24

Determine whether the statement below is true or false. Justify the answer. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent

1.7.25

Determine whether the statement below is true or false. Justify the answer. The columns of any $4 \times 5$ matrix are linearly dependent

1.7.26

Determine whether the statement below is true or false. Justify the answer. If x and y are linearly independent, and if z is in Span $\{ \mathbf{x}$,$\mathbf{y} \}$, then $\{ \mathbf{x}$,$\mathbf{y}$,$\mathbf{z} \}$ is linearly dependent

1.7.33

Suppose A is $a_{7} \times 5$ matrix. How many pivot columns must A have if its columns are linearly independent? Why?

1.7.34

Suppose A is $a_{5} \times 7$ matrix. How many pivot columns must A have if its columns span $\mathbb{R}^{5}$ ? Why?

1.7.37

Given $A = \begin{bmatrix} 4 & 1 & 3 \\ -3 & 3 & -6 \\ -8 & -1 & -7 \\ 5 & 0 & 5 \end{bmatrix}$, observe that the first column is the sum of the second and third columns. Find a nontrivial solution of A$\mathbf{x}$=$\mathbf{0}$without performing row operations. [Hint: Write A$\mathbf{x}$=$\mathbf{0}$as a vector equation.]

1.7.38

Given $A = \begin{bmatrix} 4 & 14 & 5 \\ -7 & -15 & -4 \\ -2 & -10 & -4 \end{bmatrix}$, observe that the first column + twice the third column = the second column. Find a nontrivial solution of A$\mathbf{x}$=$\mathbf{0}$without performing row operations. [Hint: Write A$\mathbf{x}$=$\mathbf{0}$as a vector equation.]

§2.1 Matrix Operations

2.1.1

Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let $A = \begin{bmatrix} 2 & 0 & -1 \\ 4 & -5 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 8 & -4 & 1 \\ 2 & -5 & -4 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 3 \\ -1 & 1 \end{bmatrix}$, and $D = \begin{bmatrix} 2 & 4 \\ -2 & 4 \end{bmatrix}$. -3 A, B -3 A, AC, CD

2.1.2

Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let $A = \begin{bmatrix} 2 & 0 & -3 \\ 3 & -4 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 8 & -5 & 1 \\ 1 & -4 & -3 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 3 \\ -3 & 2 \end{bmatrix}$, $D = \begin{bmatrix} 4 & 6 \\ -1 & 5 \end{bmatrix}$, and $E = \begin{bmatrix} -4 \\ 3 \end{bmatrix}$. $A + 3 B, 2 C - 3 E$, DB, EB

2.1.5

Compute the product AB by the definition of the product of matrices, where A$\mathbf{b}_{1}$and A$\mathbf{b}_{2}$are computed separately, and by the row-column rule for computing AB. $A = \begin{bmatrix} -2 & 4 \\ 1 & 5 \\ 6 & -3 \end{bmatrix}$, $B = \begin{bmatrix} 3 & -1 \\ -3 & 2 \end{bmatrix}$

2.1.7

If a matrix A is $6 \times 9$ and the product AB is $6 \times 7$, what is the size of B?

2.1.8

How many rows does B have if BC is $a_{5} \times 2$ matrix?

2.1.9

Let $A = \begin{bmatrix} 3 & 2 \\ -1 & 2 \end{bmatrix}$and $B = \begin{bmatrix} 2 & 8 \\ -4 & k \end{bmatrix}$. What value(s) of k, if any, will make AB = BA?

2.1.10

Let $A = \begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix}$, $B = \begin{bmatrix} 11 & 1 \\ 8 & 7 \end{bmatrix}$, and $C = \begin{bmatrix} 2 & -8 \\ 2 & 1 \end{bmatrix}$. Verify that AB = AC and yet B not = C

2.1.15

Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. If A and B are $2 \times 2$ with columns$\mathbf{a}_{1}$,$\mathbf{a}_{2}$, and$\mathbf{b}_{1}$,$\mathbf{b}_{2}$, respectively, then AB =$\begin{bmatrix} \mathbf{a}_{1} \mathbf{b}_{1} & \mathbf{a}_{2} \mathbf{b}_{2} \end{bmatrix}$

2.1.16

Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. If A and B are $3 \times 3$ matrices and $B = \begin{bmatrix} \mathbf{b}_{1} & \mathbf{b}_{2} & \mathbf{b}_{3} \end{bmatrix}$, then AB =$\begin{bmatrix} A \mathbf{b}_{1} + A \mathbf{b}_{2} + A \mathbf{b}_{3} \end{bmatrix}$

2.1.19

Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Determine whether the statement below is true or false. Justify the answer. AB + AC = A(B $+ C)$

2.1.20

Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer. $A^{Upper} T + B^{Upper} T = (A + B)^{Upper}$ T

2.1.21

Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer. (AB)C = (AC)B

2.1.22

Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. $(AB)^{Upper} T = A^{Upper} T B^{Upper}$ T

2.1.23

Determine whether the statement below is true or false. Justify the answer. The transpose of a product of matrices = the product of their transposes in the same order

2.1.24

Determine whether the statement below is true or false. Justify the answer. The transpose of a sum of matrices = the sum of their transposes

2.1.26

Suppose the first two columns,$\mathbf{b}_{1}$and$\mathbf{b}_{2}$, of B are equal. What can you say

2.1.27

Suppose the fourth column of B is the sum of the first and last columns. What can be said

2.1.47

Let $S = \begin{bmatrix} 0 & 9 & 0 & 0 & 0 \\ 0 & 0 & 9 & 0 & 0 \\ 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$. Compute $S^{k}$ for $k = 2$,…, 6

2.1.52

Use the matrix $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$to scrub the data in matrix N which contains dates of accidents at a certain airport. $N = \begin{bmatrix} 10 & 10 & 10 & 10 & 11 & 11 & 11 \\ 1 & 14 & 20 & 21 & 4 & 9 & 24 \\ 2019 & 2019 & 2019 & 2019 & 2019 & 2019 & 2019 \end{bmatrix}$

2.1.4

Compute $A - 2 I_{3}$ and ($2 I_{3}$)A, where $A = \begin{bmatrix} 6 & -2 & 4 \\ -5 & 2 & -5 \\ -3 & 1 & 2 \end{bmatrix}$

§2.2 The Inverse of a Matrix

2.2.1

Find the inverse of the matrix.$\begin{bmatrix} 4 & 6 \\ 8 & 7 \end{bmatrix}$

2.2.3

Find the inverse of the matrix.$\begin{bmatrix} 7 & 7 \\ -6 & -7 \end{bmatrix}$

2.2.4

Find the inverse of the matrix.$\begin{bmatrix} -5 & -4 \\ 3 & 2 \end{bmatrix}$

2.2.9

Let $A = \begin{bmatrix} 1 & 2 \\ 8 & 18 \end{bmatrix}$,$\mathbf{b}_{1}$=$\begin{bmatrix} 0 \\ 8 \end{bmatrix}$,$\mathbf{b}_{2}$=$\begin{bmatrix} 0 \\ -10 \end{bmatrix}$,$\mathbf{b}_{3}$=$\begin{bmatrix} 2 \\ 14 \end{bmatrix}$, and$\mathbf{b}_{4}$=$\begin{bmatrix} 4 \\ 22 \end{bmatrix}$. a. Find $A^{-1}$ and use it solve the four equations $A x = \mathbf{b}_{1}$, $A x = \mathbf{b}_{2}$, $A x = \mathbf{b}_{3}$, and $A x = \mathbf{b}_{4}$. b. The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix [A$\mathbf{b}_{1} \mathbf{b}_{2} \mathbf{b}_{3} \mathbf{b}_{4}$]

2.2.11

Determine whether the statement below is true or false. Justify the answer. In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true

2.2.12

Determine whether the statement below is true or false. Justify the answer. A product of invertible $n \times n$ matrices is invertible, and the inverse of the product is the product of their inverses in the same order

2.2.13

Determine whether the statement below is true or false. Justify the answer. If A and B are $n \times n$ and invertible, then $A^{-1} B^{-1}$ is the inverse of AB

2.2.14

Determine whether the statement below is true or false. Justify the answer. If A is invertible, then the inverse of $A^{-1}$ is A itself

2.2.15

Determine whether the statement below is true or false. Justify the answer. If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$and ab - cd not $= 0$, then A is invertible

2.2.16

Determine whether the statement below is true or false. Justify the answer. If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$and ad = bc, then A is not invertible

2.2.17

Determine whether the statement below is true or false. Justify the answer. If A is an invertible $n \times n$ matrix, then the equation A$\mathbf{x}$=$\mathbf{b}$is consistent for each$\mathbf{b}$in $\mathbb{R}^{n}$

2.2.19

Determine whether the statement below is true or false. Justify the answer. Each elementary matrix is invertible

2.2.25

Suppose A, B, and C are invertible $n \times n$ matrices. Show that ABC is also invertible by introducing a matrix D such that (ABC)D = I and D(ABC) = I

2.2.31

Explain why the columns of an $n \times n$ matrix A are linearly independent when A is invertible

2.2.32

Explain why the columns of an $n \times n$ matrix A span $\mathbb{R}^{n}$ when A is invertible

2.2.39

Find the inverse of the matrix, if it exists. Use the algorithm for finding $A^{-1}$ by row reducing A I ]. $A = \begin{bmatrix} 1 & 6 \\ 5 & 29 \end{bmatrix}$

2.2.40

Find the inverse of the matrix, if it exists. Use the algorithm for finding $A^{-1}$ by row reducing A I. $A = \begin{bmatrix} 4 & 3 \\ 2 & 7 \end{bmatrix}$

2.2.41

Find the inverse of the given matrix, if it exists. Use the algorithm for finding $A^{-1}$ by row reducing A I. $A = \begin{bmatrix} 1 & 0 & -2 \\ -3 & 1 & 4 \\ 3 & 2 & 4 \end{bmatrix}$

2.2.42

Find the inverse of the given matrix, if it exists. $A = \begin{bmatrix} 1 & -2 & 1 \\ 3 & -5 & 2 \\ -2 & 5 & -3 \end{bmatrix}$

§2.3 Characterizations of Invertible Matrices

2.3.1

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 4 & 3 \\ -3 & -6 \end{bmatrix}$

2.3.3

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 4 & 0 & 0 \\ -3 & -5 & 0 \\ 7 & 6 & -3 \end{bmatrix}$

2.3.4

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 4 & 0 & -4 \\ 2 & 0 & 5 \\ -4 & 0 & 6 \end{bmatrix}$

2.3.5

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 0 & 3 & -4 \\ 2 & 0 & 2 \\ -8 & -9 & 4 \end{bmatrix}$

2.3.6

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 1 & -4 & -6 \\ 0 & 4 & 2 \\ -4 & 13 & 0 \end{bmatrix}$

2.3.8

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.$\begin{bmatrix} 3 & 5 & 7 & 5 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & 2 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

2.3.11

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If the equation $A x = 0$ has only the trivial solution, then A is row equivalent to the $n \times n$ identity matrix

2.3.13

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If the columns of A span $\mathbb{R}^{n}$, then the columns are linearly independent

2.3.14

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If the columns of A are linearly independent, then the columns of A span $\mathbb{R}^{n}$

2.3.15

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If A is an $n \times n$ matrix, then the equation A$\mathbf{x}$=$\mathbf{b}$has at least one solution for each$\mathbf{b}$in $\mathbb{R}^{n}$

2.3.16

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If the equation $A x = b$ has at least one solution for each b in $\mathbb{R}^{n}$, then the solution is unique for each b

2.3.17

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If the equation $A x = 0$ has a nontrivial solution, then A has fewer than n pivot positions

2.3.20

For this exercise assume that the matrices are all $n \times$ n. The statement in this exercise is an implication of the form “If “statement 1”, then “statement 2”.” Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. Mark the implication as False if “statement 2” is false but “statement 1” is true. Justify your answer. If there is a b in $\mathbb{R}^{n}$ such that the equation $A x = b$ is inconsistent, then the transformation$\mathbf{x}$maps to A$\mathbf{x}$is not one-to-one

2.3.21

An $m \times n$ upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer.$\begin{bmatrix} 3 & 4 & 7 & 4 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & 2 & 8 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

2.3.30

Suppose H is an $n \times n$ matrix. If the equation $H x = c$ is inconsistent for some c in $\mathbb{R}^{n}$, what can you say

§3.1 Introduction to Determinants

3.1.1

Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.$\begin{bmatrix} 3 & 0 & 4 \\ 3 & 3 & 3 \\ 0 & 5 & -1 \end{bmatrix}$

3.1.3

Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. $\left| \begin{bmatrix} 3 & -3 & 8 \\ 8 & 5 & 3 \\ 5 & 8 & -5 \end{bmatrix} \right|$

3.1.4

Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. $\left| \begin{bmatrix} 2 & -2 & 4 \\ 3 & 1 & 3 \\ 1 & 5 & -1 \end{bmatrix} \right|$

3.1.5

Compute the determinant of the following matrix using a cofactor expansion across the first row. $\left| \begin{bmatrix} 3 & 7 & -2 \\ 2 & 0 & 6 \\ 4 & 5 & 3 \end{bmatrix} \right|$

3.1.9

Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. $\left| \begin{bmatrix} 3 & 0 & 0 & 4 \\ 2 & 7 & 2 & -2 \\ 3 & 0 & 0 & 0 \\ 1 & 2 & 1 & 7 \end{bmatrix} \right|$

3.1.10

Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of computation. $\left| \begin{bmatrix} 2 & -6 & 3 & 4 \\ 0 & 0 & 4 & 0 \\ 1 & -2 & -3 & 3 \\ 5 & 0 & 5 & 6 \end{bmatrix} \right|$

3.1.11

Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. $\left| \begin{bmatrix} 2 & 1 & -7 & 4 \\ 0 & -4 & 2 & -1 \\ 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 5 \end{bmatrix} \right|$

3.1.13

Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation.$\begin{bmatrix} 3 & 0 & -8 & 3 & -6 \\ 0 & 0 & 4 & 0 & 0 \\ 8 & 3 & -6 & 4 & -8 \\ 5 & 0 & 4 & 4 & -4 \\ 0 & 0 & 7 & -1 & 5 \end{bmatrix}$

3.1.14

Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. $\left| \begin{bmatrix} 2 & 3 & 3 & 4 & 0 \\ 9 & 0 & -3 & 1 & 0 \\ 3 & -6 & 7 & 7 & 1 \\ 3 & 0 & 0 & 0 & 0 \\ 2 & 2 & 4 & 2 & 0 \end{bmatrix} \right|$

3.1.17

The expansion of $a_{3} \times 3$ determinant can be remembered by this device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals. Add the downward diagonal products and subtract the upward products. Use this method to compute the following determinant.$\begin{bmatrix} 1 & 2 & -5 \\ -5 & 4 & 2 \\ -2 & -4 & -3 \end{bmatrix}$- minus $- +$ plus $+ a_{11} a_{12} a_{13} a_{11} a_{12} a_{21} a_{22} a_{23} a_{21} a_{22} a_{31} a_{32} a_{33} a_{31} a_{32}$

3.1.21

Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.$\begin{bmatrix} 4 & 8 \\ 9 & 2 \end{bmatrix}$,$\begin{bmatrix} 4 & 8 \\ 9 + 4 k & 2 + 8 k \end{bmatrix}$

3.1.25

Compute the determinant of the following elementary matrix.$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & d \\ 0 & 0 & 1 \end{bmatrix}$

3.1.26

Compute the determinant of the following elementary matrix.$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

3.1.29

Compute the determinant of the following elementary matrix.$\begin{bmatrix} 1 & 0 & 0 \\ 0 & z & 0 \\ 0 & 0 & 1 \end{bmatrix}$

3.1.37

Let $A = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix}$. Write 3 A. Is det (3 A) equal to 3 det (A) ?

§3.2 Properties of Determinants

3.2.1

State which property of determinants is illustrated in this equation. $\left| \begin{bmatrix} -7 & 8 & -1 \\ 21 & -3 & 6 \\ -5 & 7 & -6 \end{bmatrix} \right| =$ negative $\left| \begin{bmatrix} 21 & -3 & 6 \\ -7 & 8 & -1 \\ -5 & 7 & -6 \end{bmatrix} \right|$

3.2.2

The equation below illustrates a property of determinants. State the property.$\begin{bmatrix} 3 & -6 & 9 \\ 3 & 5 & -5 \\ 1 & 3 & 3 \end{bmatrix} = 3 \begin{bmatrix} 1 & -2 & 3 \\ 3 & 5 & -5 \\ 1 & 3 & 3 \end{bmatrix}$

3.2.5

Find the determinant by row reduction to echelon form. $\left| \begin{bmatrix} 1 & 5 & -6 \\ -1 & -4 & -5 \\ 2 & 8 & 7 \end{bmatrix} \right|$

3.2.6

Find the determinant by row reduction to echelon form. $\left| \begin{bmatrix} 5 & 5 & -5 \\ 6 & 7 & -7 \\ 2 & -3 & -5 \end{bmatrix} \right|$

3.2.12

Combine the methods of row reduction and cofactor expansion to compute the determinant. $\left| \begin{bmatrix} -1 & 4 & 8 & 0 \\ 4 & 3 & 2 & 0 \\ 4 & 4 & 6 & 6 \\ 4 & 2 & 4 & 3 \end{bmatrix} \right|$

3.2.14

Combine the methods of row reduction and cofactor expansion to compute the determinant. $\left| \begin{bmatrix} -1 & -5 & -4 & -1 \\ 0 & 4 & 8 & 0 \\ -3 & -5 & -4 & -1 \\ 6 & -4 & -4 & 0 \end{bmatrix} \right|$

3.2.21

Use determinants to find out if the matrix is invertible.$\begin{bmatrix} -5 & 0 & 1 \\ 1 & -3 & -2 \\ 0 & -5 & -3 \end{bmatrix}$

3.2.22

Use determinants to find out if the matrix is invertible.$\begin{bmatrix} -25 & -2 & 3 \\ -5 & 6 & 6 \\ 0 & -10 & -9 \end{bmatrix}$

3.2.23

Use determinants to find out if the matrix is invertible.$\begin{bmatrix} 1 & -1 & -3 & 0 \\ 0 & 1 & 5 & 4 \\ 3 & -1 & -3 & 4 \\ -1 & 2 & 8 & 5 \end{bmatrix}$

3.2.29

Let A be an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If the columns of A are linearly dependent, then det $A = 0$

3.2.33

Let A and B be $n \times n$ matrices. Determine whether the statement below is true or false. Justify the answer. det(A $+ B) =$ det $A +$ det B

3.2.34

Let A be an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. det $A^{-1} =$ (- 1)det A

3.2.42

Find a formula for det(rA) when A is an $n \times n$ matrix

3.2.43

Verify that det AB = (det A)(det B), where the matrices A and B are given below. $A = \begin{bmatrix} 2 & 0 \\ 4 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 0 \\ 5 & 10 \end{bmatrix}$

3.2.45

Let A and B be $3 \times 3$ matrices, with det $A = 2$ and det $B = -9$. Use properties of determinants to complete parts (a) through (e) below

§3.3 Cramer’s Rule; Volume

3.3.1

Use Cramer’s rule to compute the solutions of the system. $4 x_{1} + 4 x_{2} = 16 -3 x_{1} + 7 x_{2} = -42$

3.3.2

Use Cramer’s rule to compute the solution of the system. $5 x_{1} + 3 x_{2} = 13 4 x_{1} + 6 x_{2} = 14$

3.3.3

Use Cramer’s rule to compute the solutions of the system. $9 x_{1} + 7 x_{2} = 13 7 x_{1} + 7 x_{2} = 9$

3.3.5

Use Cramer’s rule to compute the solution of the system.$\begin{bmatrix} x_{1} & + & x_{2} & \, & \, & = & 3 \\ -2 x_{1} & \, & \, & + & 3 x_{3} & = & 0 \\ \, & \, & x_{2} & - & 3 x_{3} & = & 2 \end{bmatrix}$

3.3.6

Use Cramer’s rule to compute the solution of the system.$\begin{bmatrix} 11 x_{1} & + & 3 x_{2} & + & x_{3} & = & 4 \\ 2 x_{1} & \, & \, & + & 5 x_{3} & = & 4 \\ -4 x_{1} & + & 4 x_{2} & \, & \, & = & 2 \end{bmatrix}$

3.3.12

Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix. $A = \begin{bmatrix} 1 & 1 & 4 \\ -3 & 3 & 2 \\ 0 & 2 & 4 \end{bmatrix}$

3.3.13

Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix. $A = \begin{bmatrix} 4 & 6 & 3 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{bmatrix}$

3.3.15

Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix. $A = \begin{bmatrix} 2 & 0 & 0 \\ -3 & 1 & 1 \\ -3 & 3 & 1 \end{bmatrix}$

3.3.19

Find the area of the parallelogram whose vertices are listed. (0, 0), (3, 7), (8, 4), (11, 11)

3.3.21

Find the area of the parallelogram whose vertices are listed. ($- 3, -2$), (0, 5), (4, -4), (7, 3)

3.3.23

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (2, 0, -5), (1, 3, 3), and (6, 1, 0)

§4.1 Vector Spaces and Subspaces

4.1.3

Let $H = \{ \begin{bmatrix} x \\ y \end{bmatrix}$: 6 x squared $+ 4 y$ squared less than or $= 1\}$, which represents the set of points on and inside an ellipse in the xy-plane. Find two specific examples long dash two vectors, and a vector and a scalar long dash to show that H is not a subspace of $\mathbb{R}^{2}$

4.1.5

Determine if the given set is a subspace of $P_{4}$. Justify your answer. The set of all polynomials of the form $p (t) = a t^{4}$, where a is in R

4.1.7

Determine if the given set is a subspace of $P_{8}$. Justify your answer. All polynomials of degree at most 8, with positive real numbers as coefficients

4.1.8

Determine if the given set is a subspace of $P_{n}$. Justify your answer. The set of all polynomials in $P_{n}$ such that $p (0) = 0$

4.1.9

Let H be the set of all vectors of the form$\begin{bmatrix} -4 t \\ 5 t \\ t \end{bmatrix}$. Find a vector v in $\mathbb{R}^{3}$ such that $H =$ Span $\{ \mathbf{v} \}$. Why does this show that H is a subspace of $\mathbb{R}^{3}$ ?

4.1.10

Let H be the set of all vectors of the form$\begin{bmatrix} 3 t \\ 0 \\ -7 t \end{bmatrix}$. Show that H is a subspace of $\mathbb{R}^{3}$

4.1.11

Let W be the set of all vectors of the form shown on the right, where b and c are arbitrary. Find vectors u and v such that $W =$ Span $\{ \mathbf{u}$,$\mathbf{v} \}$. Why does this show that W is a subspace of $\mathbb{R}^{3}$ ?$\begin{bmatrix} 4 b - 9 c \\ -b \\ 5 c \end{bmatrix}$

4.1.12

Let W be the set of all vectors of the form$\begin{bmatrix} 2 s \\ 2 s + 4 t \\ 4 s - 4 t \\ 5 t \end{bmatrix}$. Show that W is a subspace of $\mathbb{R}^{4}$ by finding vectors u and v such that $W =$ Span{ u, v }

4.1.13

Let $v_{1} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$, $v_{2} = \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix}$, $v_{3} = \begin{bmatrix} 7 \\ 3 \\ 14 \end{bmatrix}$, and $w = \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}$. a. Is w in {$\mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3}$}? How many vectors are in {$\mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3}$}? b. How many vectors are in Span{$\mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3}$}? c. Is w in the subspace spanned by {$\mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3}$}? Why?

4.1.14

Let $v_{1} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$, $v_{2} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$, $v_{3} = \begin{bmatrix} 8 \\ 2 \\ 2 \end{bmatrix}$, and $w = \begin{bmatrix} 3 \\ 1 \\ 6 \end{bmatrix}$. Is w in the subspace spanned by {$\mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3}$}? Why?

4.1.15

Let W be the set of all vectors of the form shown on the right, where a and b represent arbitrary real numbers. Find a set S of vectors that spans W, or give an example or an explanation showing why W is not a vector space.$\begin{bmatrix} 2 a + 3 b \\ -1 \\ 2 a - 7 b \end{bmatrix}$

4.1.17

Let W be the set of all vectors of the form shown on the right, where a, b, and c represent arbitrary real numbers. Find a set S of vectors that spans W or give an example or an explanation to show that W is not a vector space.$\begin{bmatrix} 2 a + 4 b \\ 9 b - 6 c \\ 6 c - 7 a \\ 7 b \end{bmatrix}$

4.1.22

For fixed positive integers m and n, the set Upper $M_{m} \times n$ of all $m \times n$ matrices is a vector space, under the usual operations of addition of matrices and multiplication by real scalars. Let F be a fixed $3 \times 2$ matrix, and let H be the set of all matrices A in Upper $M_{2} \times 4$ with the property that FA $= 0$ (the zero matrix in Upper $M_{3} \times 4$). Determine if H is a subspace of Upper $M_{2} \times 4$

4.1.24

Determine whether the statement is True or False. Justify your answer. A vector is any element of a vector space

4.1.26

Determine whether the statement is True or False. Justify your answer. If u is a vector in a vector space V, then ($- 1) u$ is the same as the negative of u

4.1.30

Determine whether the statement is True or False. Justify your answer. $\mathbb{R}^{2}$ is a subspace of $\mathbb{R}^{3}$

4.1.34

The axioms for a vector space V can be used to prove the elementary properties for a vector space. Because of Axiom 2, Axioms 2 and 4 imply, respectively, that$\mathbf{0}$+$\mathbf{u}$=$\mathbf{u}$and -$\mathbf{u}$+$\mathbf{u}$=$\mathbf{0}$for all$\mathbf{u}$. Complete the proof that $- u$ is unique by showing that if $u + w = 0$, then $w = - u$. Use the ten axioms of a vector space to justify each step. Axioms In the following axioms,$\mathbf{u}$,$\mathbf{v}$, and$\mathbf{w}$are in vector space V and c and d are scalars. 1. The sum$\mathbf{u}$+$\mathbf{v}$is in V. 2.$\mathbf{u}$+$\mathbf{v}$=$\mathbf{v}$+$\mathbf{u}_{3}$. ($\mathbf{u}$+$\mathbf{v}$) +$\mathbf{w}$=$\mathbf{u}$+ ($\mathbf{v}$+$\mathbf{w}$) 4. V has a vector$\mathbf{0}$such that$\mathbf{u}$+$\mathbf{0}$=$\mathbf{u}$. 5. For each$\mathbf{u}$in V, there is a vector -$\mathbf{u}$in V such that$\mathbf{u}$+ ($- 1) \mathbf{u}$=$\mathbf{0}$. 6. The scalar multiple c$\mathbf{u}$is in V. 7. c($\mathbf{u}$+$\mathbf{v}$) $= c \mathbf{u} + c \mathbf{v}_{8}$. $(c + d) \mathbf{u} = c \mathbf{u} + d \mathbf{u}_{9}$. c(d$\mathbf{u}$) = (cd)$\mathbf{u}_{10}$. 1$\mathbf{u}$=$\mathbf{u}$

§4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations

4.2.1

Determine if $w = \begin{bmatrix} 5 \\ 5 \\ -1 \end{bmatrix}$is in Nul A, where $A = \begin{bmatrix} 4 & -2 & -3 \\ 6 & -2 & -5 \\ -3 & 1 & 2 \end{bmatrix}$

4.2.3

Find an explicit description of Nul A by listing vectors that span the null space. $A = \begin{bmatrix} 1 & 3 & 4 & 0 \\ 0 & 1 & 2 & -4 \end{bmatrix}$

4.2.5

Find an explicit description of Nul A by listing vectors that span the null space. $A = \begin{bmatrix} 1 & 4 & 0 & -5 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{bmatrix}$

4.2.6

Find an explicit description of Nul A by listing vectors that span the null space. $A = \begin{bmatrix} 1 & 6 & -3 & -7 & 1 \\ 0 & 1 & -8 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

4.2.8

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. $W = \{ \begin{bmatrix} r \\ s \\ t \end{bmatrix}$: $4 r - 1 = s + 8 t\}$

4.2.9

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. $W = \{ \begin{bmatrix} p \\ q \\ r \\ s \end{bmatrix}$:$\begin{bmatrix} 3 p - 3 q = -s \\ 2 p = -2 s - 3 r \end{bmatrix} \}$

4.2.11

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. $W = \{ \begin{bmatrix} s - 2 t \\ 3 + 3 s \\ 2 s + t \\ 2 s \end{bmatrix}$: $s, t real\}$

4.2.13

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. $W = \{ \begin{bmatrix} c - 2 d \\ d \\ c \end{bmatrix}$: $c, d real\}$

4.2.15

Find A such that the given set is Col A. $\{ \begin{bmatrix} 3 s + t \\ 3 r - s \\ r - 2 s + 3 t \\ -r + s + 2 t \end{bmatrix}$: $r, s, t real\}$

4.2.17

Complete parts (a) and (b) for the matrix below. $A = \begin{bmatrix} -5 & -8 & -5 & -4 \\ -1 & 9 & -8 & 1 \\ 9 & 4 & 3 & -5 \end{bmatrix}$

4.2.20

Complete parts (a) and (b) for the matrix below. $A = \begin{bmatrix} -9 & 0 & 9 \end{bmatrix}$

4.2.21

For the matrix A below, find a nonzero vector in Nul A, a nonzero vector in Col A, and a nonzero vector in Row A. $A = \begin{bmatrix} -15 & 6 \\ -5 & 2 \\ 10 & -4 \\ 15 & -6 \end{bmatrix}$

4.2.23

Let $A = \begin{bmatrix} -12 & 36 \\ -4 & 12 \end{bmatrix}$and $w = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$. Determine if w is in Col(A). Is w in Nul(A)?

4.2.24

Let $A = \begin{bmatrix} -8 & -2 & -20 \\ 8 & 6 & 28 \\ 2 & 0 & 4 \end{bmatrix}$and $w = \begin{bmatrix} 2 \\ 2 \\ -1 \end{bmatrix}$. Determine if w is in Col(A). Is w in Nul(A)?

4.2.26

A denotes an $m \times n$ matrix. Determine whether the statement is true or false. Justify your answer. A null space is a vector space

§4.3 Linearly Independent Sets; Bases

4.3.1

Describe the set $\{ \begin{bmatrix} 6 \\ 6 \\ 6 \end{bmatrix}$,$\begin{bmatrix} 6 \\ 6 \\ 0 \end{bmatrix}$,$\begin{bmatrix} 6 \\ 0 \\ 0 \end{bmatrix} \}$

4.3.2

Describe the set $\{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \}$

4.3.3

Describe the set $\{ \begin{bmatrix} 1 \\ 0 \\ -3 \end{bmatrix}$,$\begin{bmatrix} -3 \\ 1 \\ 11 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -1 \\ -8 \end{bmatrix} \}$

4.3.4

Describe the set $\{ \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -4 \\ 2 \end{bmatrix}$,$\begin{bmatrix} -8 \\ 4 \\ 4 \end{bmatrix} \}$

4.3.6

Describe the set $\{ \begin{bmatrix} 1 \\ 3 \\ -9 \end{bmatrix}$,$\begin{bmatrix} -3 \\ 4 \\ 12 \end{bmatrix} \}$

4.3.8

Describe the set $\{ \begin{bmatrix} 1 \\ -7 \\ 2 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 2 \\ -1 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -23 \\ 8 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 3 \\ -3 \end{bmatrix} \}$

4.3.9

Find a basis for the null space of the matrix$\begin{bmatrix} 1 & 0 & -7 & 4 \\ 0 & 1 & -5 & 5 \\ 7 & -10 & 1 & -22 \end{bmatrix}$

4.3.10

Find a basis for the null space of the matrix given below.$\begin{bmatrix} 1 & 1 & -3 & 1 & 3 \\ 0 & 1 & 0 & -1 & -1 \\ 0 & 0 & -6 & 0 & 6 \end{bmatrix}$

4.3.11

Find a basis for the set of vectors in $\mathbb{R}^{3}$ in the plane $x - 7 y + 4 z = 0$. [Hint: Think of the equation as a “system” of homogeneous equations.]

4.3.12

Find a basis for the set of vectors in $\mathbb{R}^{2}$ on the line $y = 24 x$

4.3.13

Assume that A is row equivalent to B. Find bases for Nul A, Col A, and Row A. $A = \begin{bmatrix} -2 & 6 & -2 & -6 \\ 2 & -9 & -4 & 2 \\ -3 & 12 & 3 & -5 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 & 7 & 7 \\ 0 & 3 & 6 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

4.3.14

Assume that A is row equivalent to B. Find bases for Nul A, Col A, and Row A. $A = \begin{bmatrix} 1 & 1 & 3 & -5 & 8 \\ 1 & 1 & 0 & 1 & 8 \\ 2 & 2 & -5 & 12 & 5 \\ 3 & 3 & 0 & 3 & 5 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 1 & 0 & 1 & 3 \\ 0 & 0 & 3 & -6 & 3 \\ 0 & 0 & 0 & 0 & -4 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

4.3.16

Find a basis for the space spanned by the given vectors.$\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}$,$\begin{bmatrix} -6 \\ 0 \\ 0 \\ 6 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -3 \\ 3 \\ -1 \end{bmatrix}$,$\begin{bmatrix} 2 \\ -12 \\ 12 \\ -1 \end{bmatrix}$,$\begin{bmatrix} -1 \\ -3 \\ 3 \\ 2 \end{bmatrix}$

4.3.19

Let$\mathbf{v}_{1}$=$\begin{bmatrix} 1 \\ -6 \\ 7 \end{bmatrix}$,$\mathbf{v}_{2}$=$\begin{bmatrix} 4 \\ 9 \\ -8 \end{bmatrix}$,$\mathbf{v}_{3}$=$\begin{bmatrix} 9 \\ 1 \\ 3 \end{bmatrix}$, and $H =$ Span $\{ \mathbf{v}_{1}$,$\mathbf{v}_{2}$,$\mathbf{v}_{3} \}$. It can be verified that 7$\mathbf{v}_{1} + 5 \mathbf{v}_{2} - 3 \mathbf{v}_{3} = 0$. Use this information to find a basis for H

4.3.22

Determine if the following statement is true or false. Justify the answer. A linearly independent set in a subspace H is a basis for H

4.3.24

Determine if the following statement is true or false. Justify the answer. If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V

4.3.26

Determine if the following statement is true or false. Justify the answer. A basis is a linearly independent set that is as large as possible

4.3.28

Determine if the following statement is true or false. Justify the answer. The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A

4.3.30

Determine if the following statement is true or false. Justify the answer. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A

4.3.32

Determine if the following statement is true or false. Justify the answer. If A and B are row equivalent, then their row spaces are the same

4.3.36

In the vector space of all real-valued functions, find a basis for the subspace spanned by $\{sine t$, sine 2 t, sine t cosine $t\}$

4.3.39

Let $S = \{ \mathbf{v}_{1}$,…, $\mathbf{v}_{k}\}$ be a set of k vectors in $\mathbb{R}^{n}$, with k less than n. Use a theorem

4.3.44

Consider the polynomials$\mathbf{p}_{1} (t) = 4 + 3 t$,$\mathbf{p}_{2} (t) = 4 - 3 t$, and$\mathbf{p}_{3} (t) = 8$ (for all t). By inspection, write a linear dependence relation among$\mathbf{p}_{1}$,$\mathbf{p}_{2}$, and$\mathbf{p}_{3}$. Then find a basis for Span $\{ \mathbf{p}_{1}$,$\mathbf{p}_{2}$,$\mathbf{p}_{3} \}$

§4.5 The Dimension of a Vector Space

4.5.6

For the subspace below, (a) find a basis, and (b) state the dimension. $\{ \begin{bmatrix} 6 a + 12 b - 2 c \\ 9 a - 3 b - 3 c \\ -12 a + 5 b + 4 c \\ -3 a + b + c \end{bmatrix}$: a, b, c in $R\}$

4.5.8

For the subspace below, (a) find a basis for the subspace, and (b) state the dimension. $\{(a, b, c, d)$ : $a - 5 b + 6 c = 0\}$

4.5.9

Find the dimension of the subspace spanned by the given vectors.$\begin{bmatrix} 1 \\ 0 \\ -2 \end{bmatrix}$,$\begin{bmatrix} 2 \\ 1 \\ -3 \end{bmatrix}$,$\begin{bmatrix} 5 \\ 4 \\ -6 \end{bmatrix}$,$\begin{bmatrix} -4 \\ -3 \\ 5 \end{bmatrix}$

4.5.10

Find the dimension of the subspace spanned by the given vectors.$\begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}$,$\begin{bmatrix} 3 \\ 10 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 22 \\ 71 \\ 5 \end{bmatrix}$,$\begin{bmatrix} 3 \\ 2 \\ 8 \end{bmatrix}$

4.5.11

Determine the dimensions of Nul A, Col A, and Row A for the given matrix. $A = \begin{bmatrix} 1 & 8 & -2 & -6 & 1 \\ 0 & 1 & -1 & -2 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

4.5.12

Determine the dimensions of Nul A, Col A, and Row A for the given matrix. $A = \begin{bmatrix} 1 & 4 & -6 & 2 & -3 & 6 & 0 \\ 0 & 0 & 1 & -4 & 5 & 3 & -5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

4.5.13

Determine the dimensions of Nul A, Col A, and Row A for the given matrix. $A = \begin{bmatrix} 1 & 4 & 0 & 5 \\ 0 & 1 & 0 & -2 \end{bmatrix}$

4.5.15

Determine the dimensions of Nul A, Col A, and Row A for the given matrix. $A = \begin{bmatrix} 1 & -5 & 0 \\ 0 & 1 & 7 \\ 0 & 0 & 1 \end{bmatrix}$

4.5.18

Let A be an $m \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. The number of variables in the equation $A x = 0 =$ the nullity of A

4.5.23

Let A be an $m \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A

4.5.24

Let A be an $m \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. The nullity of A is the number of columns of A that are not pivot columns

4.5.37

If the nullity of $a_{4} \times 6$ matrix A is 3, what are the dimensions of the column and row spaces of A?

§5.1 Eigenvectors and Eigenvalues

5.1.2

Is lambda $= -4$ an eigenvalue of $A = \begin{bmatrix} -9 & 5 \\ -2 & -2 \end{bmatrix}$? Why or why not?

5.1.4

Is $v = \begin{bmatrix} -2 \\ 1 \end{bmatrix}$an eigenvector of $A = \begin{bmatrix} -1 & 4 \\ 3 & 3 \end{bmatrix}$? If so, find the eigenvalue

5.1.6

Is $v = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}$an eigenvector of $A = \begin{bmatrix} -4 & 3 & 3 \\ 2 & -3 & -2 \\ -1 & 0 & -2 \end{bmatrix}$? If so, find the eigenvalue

5.1.7

Is lambda $= 8$ an eigenvalue of$\begin{bmatrix} 7 & 0 & -3 \\ 2 & 7 & 5 \\ -3 & 4 & 3 \end{bmatrix}$? If so, find one corresponding eigenvector

5.1.8

Is lambda $= 8$ an eigenvalue of$\begin{bmatrix} 6 & 2 & 2 \\ 2 & 3 & 4 \\ 0 & 1 & 6 \end{bmatrix}$? If so, find one corresponding eigenvector

5.1.9

Find a basis for the eigenspace corresponding to each listed eigenvalue of A below. $A = \begin{bmatrix} 1 & 0 \\ -2 & 3 \end{bmatrix}$, lambda $= 3, 1$

5.1.11

Find a basis for the eigenspace corresponding to the eigenvalue. $A = \begin{bmatrix} 2 & -6 \\ -4 & 7 \end{bmatrix}$, lambda $= 10$

5.1.12

Find a basis for the eigenspace corresponding to each listed eigenvalue. $A = \begin{bmatrix} 7 & 4 \\ -3 & -1 \end{bmatrix}$, lambda $= 1, 5$

5.1.13

Find a basis for the eigenspace corresponding to each listed eigenvalue of A below. $A = \begin{bmatrix} 3 & 1 & 0 \\ -2 & 0 & 0 \\ -4 & 2 & 5 \end{bmatrix}$, lambda $= 5, 1, 2$

5.1.14

Find a basis for the eigenspace corresponding to the eigenvalue of A given below. $A = \begin{bmatrix} 6 & 0 & -2 \\ 3 & 0 & -16 \\ -2 & 1 & 11 \end{bmatrix}$, lambda $= 5$

5.1.15

Find a basis for the eigenspace corresponding to the eigenvalue. $A = \begin{bmatrix} 4 & 1 & 1 \\ 4 & 7 & 4 \\ 3 & 3 & 6 \end{bmatrix}$, lambda $= 3$

5.1.16

Find a basis for the eigenspace corresponding to the eigenvalue of A given below. $A = \begin{bmatrix} 4 & 0 & 2 & 0 \\ 3 & 2 & 5 & 0 \\ 4 & -3 & 8 & 0 \\ 5 & -2 & 8 & 3 \end{bmatrix}$, lambda $= 3$

5.1.17

Find the eigenvalues of the matrix.$\begin{bmatrix} 0 & 0 & 0 \\ 0 & -8 & 9 \\ 0 & 0 & -4 \end{bmatrix}$

5.1.22

A is an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If A$\mathbf{x}$= lambda$\mathbf{x}$for some scalar lambda, then$\mathbf{x}$is an eigenvector of A

§5.2 The Characteristic Equation

5.2.1

Find the characteristic polynomial and the eigenvalues of the matrix.$\begin{bmatrix} 9 & 4 \\ 4 & 9 \end{bmatrix}$

5.2.3

Find the characteristic polynomial and the eigenvalues of the matrix.$\begin{bmatrix} -3 & 6 \\ 1 & -1 \end{bmatrix}$

5.2.4

Find the characteristic polynomial and the eigenvalues of the matrix.$\begin{bmatrix} 6 & -2 \\ -5 & 2 \end{bmatrix}$

5.2.5

Find the characteristic polynomial and the eigenvalues of the matrix.$\begin{bmatrix} -9 & 1 \\ -1 & -7 \end{bmatrix}$

5.2.6

Find the characteristic polynomial and the eigenvalues of the matrix.$\begin{bmatrix} 6 & 2 \\ -2 & 5 \end{bmatrix}$

5.2.9

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} 1 & 0 & -1 \\ 3 & 4 & -2 \\ 0 & 5 & 0 \end{bmatrix}$

5.2.10

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} 0 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 0 \end{bmatrix}$

5.2.11

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} 3 & 0 & 0 \\ -8 & 5 & -3 \\ -6 & 0 & 2 \end{bmatrix}$

5.2.12

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} -2 & 0 & -8 \\ -8 & 5 & -6 \\ 0 & 0 & 6 \end{bmatrix}$

5.2.13

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} 5 & -3 & 0 \\ -3 & 9 & 0 \\ 5 & 6 & 4 \end{bmatrix}$

5.2.14

Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants. [Note: Finding the characteristic polynomial of $a_{3} \times 3$ matrix is not easy to do with just row operations, because the variable lambda is involved.]$\begin{bmatrix} 6 & 9 & 3 \\ 0 & 3 & 0 \\ 7 & 6 & -4 \end{bmatrix}$

5.2.15

For the matrix, list the real eigenvalues, repeated according to their multiplicities.$\begin{bmatrix} 9 & -3 & 0 & 6 \\ 0 & 6 & 2 & -3 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 9 \end{bmatrix}$

5.2.16

For the matrix, list the eigenvalues, repeated according to their multiplicities.$\begin{bmatrix} 3 & 0 & 0 & 0 \\ -8 & 3 & 0 & 0 \\ 0 & -7 & -8 & 0 \\ 5 & -4 & -7 & 9 \end{bmatrix}$

5.2.18

It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace corresponding to lambda. Find h in the matrix A below such that the eigenspace for lambda $= 5$ is two-dimensional. $A = \begin{bmatrix} 5 & -2 & 8 & 5 \\ 0 & 3 & h & 0 \\ 0 & 0 & 5 & 2 \\ 0 & 0 & 0 & -2 \end{bmatrix}$

5.2.27

Let A be an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If lambda $+ 5$ is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A

§5.3 Diagonalization

5.3.11

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.$\begin{bmatrix} 5 & 7 & -10 \\ 2 & 12 & -12 \\ 2 & 8 & -8 \end{bmatrix}$; lambda $= 2, 3, 4$

5.3.12

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.$\begin{bmatrix} 5 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 5 \end{bmatrix}$; lambda $= 3, 9$

5.3.15

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.$\begin{bmatrix} -2 & 1 & 1 \\ -4 & 3 & 4 \\ -2 & 2 & 1 \end{bmatrix}$; lambda $= - 1, 4$

5.3.16

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.$\begin{bmatrix} 0 & -4 & -6 \\ -1 & 0 & -3 \\ 1 & 2 & 5 \end{bmatrix}$; lambda $= 1, 2$

5.3.17

Diagonalize the following matrix.$\begin{bmatrix} 4 & 0 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix}$

5.3.18

Diagonalize the following matrix. One eigenvalue is lambda $= 5$ and one eigenvector is (-4, 3, 2).$\begin{bmatrix} -19 & -36 & 24 \\ 18 & 32 & -18 \\ 12 & 18 & -7 \end{bmatrix}$

5.3.19

Diagonalize the following matrix.$\begin{bmatrix} 7 & -5 & 0 & 5 \\ 0 & 3 & 1 & -2 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix}$

5.3.21

Let A, P, and D be $n \times n$ matrices. Determine whether the statement below is true or false. Justify the answer. A is diagonalizable if $A = PDP^{-1}$ for some matrix D and some invertible matrix P

5.3.22

Let A be an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If $\mathbb{R}^{n}$ has a basis of eigenvectors of A, then A is diagonalizable

5.3.26

Let A be an $n \times n$ matrix. Determine whether the statement below is true or false. Justify the answer. If A is diagonalizable, then A has n distinct eigenvalues

5.3.27

Assume A, P, and D are $n \times n$ matrices. Determine whether the statement below is true or false. Justify the answer. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A

5.3.30

A is $a_{3} \times 3$ matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?

5.3.1

Let $A = PDP^{-1}$ and P and D as shown below. Compute $A^{4}$. $P = \begin{bmatrix} 1 & 4 \\ 2 & 7 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}$

5.3.2

Let $A = PDP^{-1}$ and P and D as shown below. Compute $A^{4}$. $P = \begin{bmatrix} 2 & -3 \\ -3 & 5 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 \\ 0 & one half \end{bmatrix}$

5.3.3

Use the factorization $A = PDP^{-1}$ to compute $A^{k}$, where k represents an arbitrary integer.$\begin{bmatrix} a & 5 (b - a) \\ 0 & b \end{bmatrix}$=$\begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \begin{bmatrix} 1 & -5 \\ 0 & 1 \end{bmatrix}$

§6.1 Inner Product, Length, and Orthogonality

6.1.1

Compute$\mathbf{u} \times \mathbf{u}$,$\mathbf{v} \times \mathbf{u}$, and $\frac{ \mathbf{v} \times \mathbf{u}$}{$\mathbf{u} \times \mathbf{u}$} using the vectors$\mathbf{u}$=$\begin{bmatrix} -3 \\ 5 \end{bmatrix}$and$\mathbf{v}$=$\begin{bmatrix} 7 \\ 4 \end{bmatrix}$

6.1.2

Compute$\mathbf{w} \times \mathbf{w}$,$\mathbf{x} \times \mathbf{w}$, and $\frac{ \mathbf{x} \times \mathbf{w}$}{$\mathbf{w} \times \mathbf{w}$} using the vectors$\mathbf{w}$=$\begin{bmatrix} 3 \\ -4 \\ -5 \end{bmatrix}$and$\mathbf{x}$=$\begin{bmatrix} 5 \\ -3 \\ 1 \end{bmatrix}$

6.1.5

Compute the quantity using the vectors$\mathbf{u}$=$\begin{bmatrix} -5 \\ 2 \end{bmatrix}$, and$\mathbf{v}$=$\begin{bmatrix} 4 \\ 5 \end{bmatrix}$. $(\frac{ \mathbf{u} \times \mathbf{v}$}{$\mathbf{v} \times \mathbf{v}$})$\mathbf{v}$

6.1.7

Compute $\lVert \mathbf{w} \rVert$ using$\mathbf{w}$=$\begin{bmatrix} 5 \\ 3 \\ -2 \end{bmatrix}$

6.1.9

Find a unit vector in the direction of the given vector.$\begin{bmatrix} 27 \\ -36 \end{bmatrix}$

6.1.10

Find a unit vector in the direction of the given vector.$\begin{bmatrix} -3 \\ 27 \\ -3 \end{bmatrix}$

6.1.14

Find the distance between $u = \begin{bmatrix} 0 \\ -6 \\ 2 \end{bmatrix}$and $z = \begin{bmatrix} -5 \\ -1 \\ 6 \end{bmatrix}$

6.1.15

Determine if the following vectors are orthogonal. $a = \begin{bmatrix} 8 \\ -7 \end{bmatrix}$, $b = \begin{bmatrix} -2 \\ -5 \end{bmatrix}$

6.1.16

Determine if the following vectors are orthogonal. $u = \begin{bmatrix} 12 \\ 5 \\ 1 \end{bmatrix}$, $v = \begin{bmatrix} 1 \\ -5 \\ 13 \end{bmatrix}$

6.1.17

Determine if the following vectors are orthogonal. $u = \begin{bmatrix} 8 \\ 2 \\ -7 \\ 0 \end{bmatrix}$, $v = \begin{bmatrix} -4 \\ 9 \\ -2 \\ 4 \end{bmatrix}$

6.1.18

Determine if the following vectors are orthogonal. $u = \begin{bmatrix} -2 \\ 7 \\ 5 \\ 0 \end{bmatrix}$, $v = \begin{bmatrix} 1 \\ -6 \\ 19 \\ -5 \end{bmatrix}$

6.1.19

Determine whether the statement below is true or false. Justify the answer. The vector is in $\mathbb{R}^{n}$.$\mathbf{v} \times \mathbf{v} = \lVert \mathbf{v} \rVert$ squared

§6.2 Orthogonal Sets

6.2.2

Determine whether the set of vectors is orthogonal.$\begin{bmatrix} 1 \\ -7 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 1 \\ 7 \end{bmatrix}$,$\begin{bmatrix} -50 \\ -7 \\ 1 \end{bmatrix}$

6.2.3

Determine whether the set of vectors is orthogonal.$\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$,$\begin{bmatrix} 0 \\ 1 \\ -2 \end{bmatrix}$,$\begin{bmatrix} -5 \\ 2 \\ 1 \end{bmatrix}$

6.2.5

Determine whether the set of vectors is orthogonal.$\begin{bmatrix} 6 \\ -3 \\ 1 \\ 6 \end{bmatrix}$,$\begin{bmatrix} -1 \\ 6 \\ -6 \\ 5 \end{bmatrix}$,$\begin{bmatrix} 12 \\ 35 \\ 33 \\ 0 \end{bmatrix}$

6.2.6

Determine whether the set of vectors is orthogonal.$\begin{bmatrix} 4 \\ -5 \\ 0 \\ 3 \end{bmatrix}$,$\begin{bmatrix} -5 \\ 2 \\ -3 \\ 10 \end{bmatrix}$,$\begin{bmatrix} 3 \\ 3 \\ 4 \\ 1 \end{bmatrix}$

6.2.7

Show that $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$ is an orthogonal basis for $\mathbb{R}^{2}$. Then express x as a linear combination of the u ’s.$\mathbf{u}_{1}$=$\begin{bmatrix} 6 \\ -8 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} 16 \\ 12 \end{bmatrix}$, and $x = \begin{bmatrix} 7 \\ -1 \end{bmatrix}$

6.2.10

Show that $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2}$,$\mathbf{u}_{3} \}$ is an orthogonal basis for $\mathbb{R}^{3}$. Then express x as a linear combination of the u ’s.$\mathbf{u}_{1}$=$\begin{bmatrix} 4 \\ -4 \\ 0 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} 2 \\ 2 \\ -1 \end{bmatrix}$,$\mathbf{u}_{3}$=$\begin{bmatrix} 1 \\ 1 \\ 4 \end{bmatrix}$, and $x = \begin{bmatrix} 5 \\ -2 \\ 1 \end{bmatrix}$

6.2.11

Compute the orthogonal projection of$\begin{bmatrix} 1 \\ 7 \end{bmatrix}$onto the line through$\begin{bmatrix} -8 \\ 4 \end{bmatrix}$and the origin

6.2.13

Let$\mathbf{y}$=$\begin{bmatrix} 1 \\ 4 \end{bmatrix}$and$\mathbf{u}$=$\begin{bmatrix} 7 \\ -6 \end{bmatrix}$. Write$\mathbf{y}$as the sum of two orthogonal vectors, one in Span $\{ \mathbf{u} \}$ and one orthogonal to$\mathbf{u}$

6.2.14

Let$\mathbf{y}$=$\begin{bmatrix} 1 \\ 2 \end{bmatrix}$and$\mathbf{u}$=$\begin{bmatrix} 4 \\ 3 \end{bmatrix}$. Write$\mathbf{y}$as the sum of a vector in Span $\{ \mathbf{u} \}$ and a vector orthogonal to$\mathbf{u}$

6.2.17

Determine whether the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.$\mathbf{u}_{1}$=$\begin{bmatrix} one third \\ two thirds \\ one third \end{bmatrix}$and$\mathbf{u}_{2}$=$\begin{bmatrix} one half \\ 0 \\ -one half \end{bmatrix}$

6.2.19

Determine if the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.$\mathbf{u}$=$\begin{bmatrix} -0.8 \\ -0.6 \end{bmatrix}$,$\mathbf{v}$=$\begin{bmatrix} -0.6 \\ 0.8 \end{bmatrix}$

6.2.21

Determine whether the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.$\mathbf{u}_{1}$=$\begin{bmatrix} \frac{1}{\sqrt{226}} \\ \frac{15}{\sqrt{452}} \\ \frac{15}{\sqrt{452}} \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} \frac{15}{\sqrt{226}} \\ -\frac{1}{\sqrt{452}} \\ -\frac{1}{\sqrt{452}} \end{bmatrix}$,$\mathbf{u}_{3}$=$\begin{bmatrix} 0 \\ -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}$

§6.3 Orthogonal Projections

6.3.4

Verify that $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$ is an orthogonal set, and then find the orthogonal projection of y onto Span $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$. $y = \begin{bmatrix} 6 \\ 2 \\ -5 \end{bmatrix}$,$\mathbf{u}_{1}$=$\begin{bmatrix} 5 \\ 6 \\ 0 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} -6 \\ 5 \\ 0 \end{bmatrix}$

6.3.5

Verify that $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$ is an orthogonal set, and then find the orthogonal projection of y onto Span $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$. $y = \begin{bmatrix} 1 \\ -2 \\ 4 \end{bmatrix}$,$\mathbf{u}_{1}$=$\begin{bmatrix} 5 \\ 1 \\ -2 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} -1 \\ 1 \\ -2 \end{bmatrix}$

6.3.8

Let W be the subspace spanned by$\mathbf{u}_{1}$and$\mathbf{u}_{2}$, and write$\mathbf{y}$as the sum of a vector in W and a vector orthogonal to W.$\mathbf{y}$=$\begin{bmatrix} -3 \\ 5 \\ 4 \end{bmatrix}$,$\mathbf{u}_{1}$=$\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} -2 \\ 5 \\ -3 \end{bmatrix}$

6.3.9

Let W be a subspace spanned by the$\mathbf{u}$’s, and write$\mathbf{y}$as the sum of a vector in W and a vector orthogonal to W.$\mathbf{y}$=$\begin{bmatrix} 2 \\ 5 \\ 5 \\ -1 \end{bmatrix}$,$\mathbf{u}_{1}$=$\begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix}$,$\mathbf{u}_{2}$=$\begin{bmatrix} -1 \\ 5 \\ 3 \\ -4 \end{bmatrix}$,$\mathbf{u}_{3}$=$\begin{bmatrix} -1 \\ 0 \\ 1 \\ 1 \end{bmatrix}$

6.3.21

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in $\mathbb{R}^{n}$. If$\mathbf{z}$is orthogonal to$\mathbf{u}_{1}$and$\mathbf{u}_{2}$and if $W =$ Span $\{ \mathbf{u}_{1}$,$\mathbf{u}_{2} \}$, then$\mathbf{z}$must be in $W^{orthogonal}$

6.3.22

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in $\mathbb{R}^{n}$. For each$\mathbf{y}$and each subspace W, the vector$\mathbf{y}$- pro $j_{Upper}$ W$\mathbf{y}$is orthogonal to W

6.3.23

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in $\mathbb{R}^{n}$. The orthogonal projection ModifyingAbove$\mathbf{y}$with caret of$\mathbf{y}$onto a subspace W can sometimes depend on the orthogonal basis for W used to compute ModifyingAbove$\mathbf{y}$with caret

6.3.26

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in $\mathbb{R}^{n}$. If W is a subspace of $\mathbb{R}^{n}$ and if$\mathbf{v}$is in both W and $W^{orthogonal}$, then$\mathbf{v}$must be the zero vector

6.3.28

Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in $\mathbb{R}^{n}$. If$\mathbf{y}$=$\mathbf{z}_{1}$+$\mathbf{z}_{2}$, where$\mathbf{z}_{1}$is in a subspace W and$\mathbf{z}_{2}$is in $W^{orthogonal}$, then$\mathbf{z}_{1}$must be the orthogonal projection of$\mathbf{y}$onto W