Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 4

Vector Spaces - all with Video Answers

Educators

VU

Section 2

Null Spaces, Column Spaces, and Linear Transformations

Problem 1

Determine if $\mathbf{w}=\left[\begin{array}{r}{1} \\ {3} \\ {-4}\end{array}\right]$ is in Nul $A,$ where
$$
A=\left[\begin{array}{rrr}{3} & {-5} & {-3} \\ {6} & {-2} & {0} \\ {-8} & {4} & {1}\end{array}\right]
$$

Michael J.

Michael J.

Numerade Educator

Problem 2

Determine if $\mathbf{w}=\left[\begin{array}{r}{5} \\ {-3} \\ {2}\end{array}\right]$ is in Nul $A,$ where
$$
A=\left[\begin{array}{rrr}{5} & {21} & {19} \\ {13} & {23} & {2} \\ {8} & {14} & {1}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 2

Let $A=\left[\begin{array}{cc}{-6} & {12} \\ {-3} & {6}\end{array}\right]$ and $\mathbf{w}=\left[\begin{array}{l}{2} \\ {1}\end{array}\right] .$ Determine if $\mathbf{w}$ is in Col $A .$ Is $\mathbf{w}$ in Nul $A$ ?

Doruk I.

Numerade Educator

Problem 3

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{llll}{1} & {3} & {5} & {0} \\ {0} & {1} & {4} & {-2}\end{array}\right]
$$

Michael J.

Numerade Educator

Problem 4

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{rrrr}{1} & {-6} & {4} & {0} \\ {0} & {0} & {2} & {0}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 5

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{rrrrr}{1} & {-2} & {0} & {4} & {0} \\ {0} & {0} & {1} & {-9} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right]
$$

OZ

Oumeng Z.

Numerade Educator

Problem 6

In Exercises $3-6,$ find an explicit description of Nul $A$ by listing vectors that span the null space.
$$
A=\left[\begin{array}{rrrrr}{1} & {5} & {-4} & {-3} & {1} \\ {0} & {1} & {-2} & {1} & {0} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 7

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{l}{a} \\ {b} \\ {c}\end{array}\right] : a+b+c=2\right\}
$$

Michael J.

Numerade Educator

Problem 8

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{l}{r} \\ {s} \\ {t}\end{array}\right] : 5 r-1=s+2 t\right\}
$$

Doruk I.

Numerade Educator

Problem 9

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{l}{a} \\ {b} \\ {c} \\ {d}\end{array}\right] : \begin{array}{l}{a-2 b=4 c} \\ {2 a=c+3 d}\end{array}\right\}
$$

Michael J.

Numerade Educator

Problem 10

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{l}{a} \\ {b} \\ {c} \\ {d}\end{array}\right] : \begin{array}{l}{a+3 b=c} \\ {b+c+a=d}\end{array}\right\}
$$

Doruk I.

Numerade Educator

Problem 11

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{c}{b-2 d} \\ {5+d} \\ {b+3 d} \\ {d}\end{array}\right] : b, d \text { real }\right\}
$$

Michael J.

Numerade Educator

Problem 12

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{c}{b-5 d} \\ {2 b} \\ {2 d+1} \\ {d}\end{array}\right] : b, d \text { real }\right\}
$$

Doruk I.

Numerade Educator

Problem 13

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{c}{c-6 d} \\ {d} \\ {c}\end{array}\right] : c, d \text { real }\right\}
$$

Michael J.

Numerade Educator

Problem 14

In Exercises $7-14$ , cither use an appropriate theorem to show that the given set, $W,$ is a vector space, or find a specific example to the contrary.
$$
\left\{\left[\begin{array}{c}{-a+2 b} \\ {a-2 b} \\ {3 a-6 b}\end{array}\right] : a, b \text { real }\right\}
$$

Doruk I.

Numerade Educator

Problem 15

In Exercises 15 and $16,$ find $A$ such that the given set is $\operatorname{Col} A$
$$
\left\{\left[\begin{array}{c}{2 s+3 t} \\ {r+s-2 t} \\ {4 r+s} \\ {3 r-s-t}\end{array}\right] : r, s, t \text { real }\right\}
$$

Michael J.

Numerade Educator

Problem 16

In Exercises 15 and $16,$ find $A$ such that the given set is $\operatorname{Col} A$
$$
\left\{\left[\begin{array}{c}{b-c} \\ {2 b+c+d} \\ {5 c-4 d} \\ {d}\end{array}\right] : b, c, d \text { real }\right\}
$$

Doruk I.

Numerade Educator

Problem 17

For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a subspace of $\mathbb{R}^{k},$ and $(b)$ find $k$ such that $\operatorname{Col} A$ is a subspace of $\mathbb{R}^{k} .$
$$
A=\left[\begin{array}{rr}{2} & {-6} \\ {-1} & {3} \\ {-4} & {12} \\ {3} & {-9}\end{array}\right]
$$

Michael J.

Numerade Educator

Problem 18

For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a subspace of $\mathbb{R}^{k},$ and $(b)$ find $k$ such that $\operatorname{Col} A$ is a subspace of $\mathbb{R}^{k} .$
$$
A=\left[\begin{array}{rrr}{7} & {-2} & {0} \\ {-2} & {0} & {-5} \\ {0} & {-5} & {7} \\ {-5} & {7} & {-2}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 19

For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a subspace of $\mathbb{R}^{k},$ and $(b)$ find $k$ such that $\operatorname{Col} A$ is a subspace of $\mathbb{R}^{k} .$
$$
A=\left[\begin{array}{rrrrr}{4} & {5} & {-2} & {6} & {0} \\ {1} & {1} & {0} & {1} & {0}\end{array}\right]
$$

Check back soon!

Problem 20

For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a subspace of $\mathbb{R}^{k},$ and $(b)$ find $k$ such that $\operatorname{Col} A$ is a subspace of $\mathbb{R}^{k} .$
$$
A=\left[\begin{array}{lllll}{1} & {-3} & {9} & {0} & {-5}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 21

With $A$ as in Exercise 17 , find a nonzero vector in Nul $A$ and a nonzero vector in $\operatorname{Col} A .$

Michael J.

Numerade Educator

Problem 22

With $A$ as in Exercise $3,$ find a nonzero vector in Nul $A$ and a nonzero vector in $\operatorname{Col} A .$

Doruk I.

Numerade Educator

Problem 24

Let $A=\left[\begin{array}{rrr}{-8} & {-2} & {-9} \\ {6} & {4} & {8} \\ {4} & {0} & {4}\end{array}\right]$ and $\mathbf{w}=\left[\begin{array}{r}{2} \\ {1} \\ {-2}\end{array}\right] .$ Determine if $\mathbf{w}$ is in $\operatorname{Col} A .$ Is $\mathbf{w}$ in Nul $A ?$

Doruk I.

Numerade Educator

Problem 25

In Exercises 25 and $26, A$ denotes an $m \times n$ matrix. Mark each statement True or False. Justify each answer.
a. The null space of $A$ is the solution set of the equation $A \mathbf{x}=\mathbf{0} .$
b. The null space of an $m \times n$ matrix is in $\mathbb{R}^{m}$ .
c. The column space of $A$ is the range of the mapping $\quad \mathbf{x} \mapsto A \mathbf{x}$
d. If the equation $A \mathbf{x}=\mathbf{b}$ is consistent, then $\operatorname{Col} A$ is $\mathbb{R}^{m} .$
e. The kernel of a linear transformation is a vector space.
f. $\mathrm{Col} A$ is the set of all vectors that can be written as $A \mathbf{x}$ for some $\mathbf{x} .$

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Problem 26

In Exercises 25 and $26, A$ denotes an $m \times n$ matrix. Mark each statement True or False. Justify each answer.
a. A null space is a vector space.
b. The column space of an $m \times n$ matrix is in $\mathbb{R}^{m}$ .
c. Col $A$ is the set of all solutions of $A \mathbf{x}=\mathbf{b}$ .
d. Nul $A$ is the kernel of the mapping $\mathbf{X} \mapsto A \mathbf{x}$
e. The range of a linear transformation is a vector space.
f. The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.

Doruk I.

Numerade Educator

Problem 27

It can be shown that a solution of the system below is $x_{1}=3$ $x_{2}=2,$ and $x_{3}=-1 .$ Use this fact and the theory from this section to explain why another solution is $x_{1}=30, x_{2}=20$ , and $x_{3}=-10$ . (Observe how the solutions are related, but make no other calculations.)
$$
\begin{aligned} x_{1}-3 x_{2}-3 x_{3} &=0 \\-2 x_{1}+4 x_{2}+2 x_{3} &=0 \\-x_{1}+5 x_{2}+7 x_{3} &=0 \end{aligned}
$$

Michael J.

Numerade Educator

Problem 28

Consider the following two systems of equations:
$$
\begin{array}{rlrl}{5 x_{1}+x_{2}-3 x_{3}} & {=0} & {5 x_{1}+x_{2}-3 x_{3}} & {=0} \\ {-9 x_{1}+2 x_{2}+5 x_{3}} & {=1} & {-9 x_{1}+2 x_{2}+5 x_{3}} & {=5} \\ {4 x_{1}+x_{2}-6 x_{3}} & {=9} & {4 x_{1}+x_{2}-6 x_{3}} & {=45}\end{array}
$$
It can be shown that the first system has a solution. Use this fact and the theory from this section to explain why the second system must also have a solution. Make no row operations.)

Doruk I.

Numerade Educator

Problem 29

Prove Theorem 3 as follows: Given an $m \times n$ matrix $A,$ an element in $\operatorname{Col} A$ has the form $A \mathbf{x}$ for some $\mathbf{x}$ in $\mathbb{R}^{n} .$ Let $A \mathbf{x}$ and $A \mathbf{w}$ represent any two vectors in $\operatorname{Col} A$
a. Explain why the zero vector is in Col $A$ .
b. Show that the vector $A \mathbf{x}+A \mathbf{w}$ is in Col $A$
c. Given a scalar $c,$ show that $c(A \mathbf{x})$ is in $\operatorname{Col} A$

Michael J.

Numerade Educator

Problem 30

Let $T : V \rightarrow W$ be a linear transformation from a vector space $V$ into a vector space $W .$ Prove that the range of $T$ is a subspace of $W .$ IHint: Typical elements of the range have the form $T(\mathbf{x})$ and $T(\mathbf{w})$ for some $\mathbf{x}, \mathbf{w}$ in $V . ]$

Doruk I.

Numerade Educator

Problem 31

Define $T : \mathbb{P}_{2} \rightarrow \mathbb{R}^{2}$ by $T(\mathbf{p})=\left[\begin{array}{c}{\mathbf{p}(0)} \\ {\mathbf{p}(1)}\end{array}\right] .$ For instance, if $\mathbf{p}(t)=3+5 t+7 t^{2},$ then $T(\mathbf{p})=\left[\begin{array}{c}{3} \\ {15}\end{array}\right]$
a. Show that $T$ is a linear transformation. [Hint: For arbitrary polynomials $\mathbf{p}, \mathbf{q}$ in $\mathbb{P}_{2}$ , compute $T(\mathbf{p}+\mathbf{q})$ and $T(c \mathbf{p}) . ]$
b. Find a polynomial $\mathbf{p}$ in $\mathbb{P}_{2}$ that spans the kernel of $T,$ and describe the range of $T$

Michael J.

Numerade Educator

Problem 32

Define $\quad$ a linear transformation $\quad T : \mathbb{P}_{2} \rightarrow \mathbb{R}^{2} \quad$ by
$T(\mathbf{p})=\left[\begin{array}{l}{\mathbf{p}(0)} \\ {\mathbf{p}(0)}\end{array}\right] .$ Find polynomials $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ in $\mathbb{P}_{2}$ that span the kernel of $T,$ and describe the range of $T$

Doruk I.

Numerade Educator

Problem 33

Let $M_{2 \times 2}$ be the vector space of all $2 \times 2$ matrices, and define $T : M_{2 \times 2} \rightarrow M_{2 \times 2}$ by $T(A)=A+A^{T},$ where $A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$
a. Show that $T$ is a linear transformation.
b. Let $B$ be any element of $M_{2 \times 2}$ such that $B^{T}=B .$ Find an $A$ in $M_{2 \times 2}$ such that $T(A)=B$ .
c. Show that the range of $T$ is the set of $B$ in $M_{2 \times 2}$ with the property that $B^{T}=B$ .
d. Describe the kernel of $T$

Michael J.

Numerade Educator

Problem 34

(Calculus required) Define $T : C[0,1] \rightarrow C[0,1]$ as follows: For $\mathbf{f}$ in $C[0,1],$ let $T(\mathbf{f})$ be the antiderivative $\mathbf{F}$ of $\mathbf{f}$ such that $\mathbf{F}(0)=0 .$ Show that $T$ is a linear transformation, and describe the kernel of $T .$ (See the notation in Exercise 20 of
Section $4.1 . )$

Doruk I.

Numerade Educator

Problem 35

Let $V$ and $W$ be vector spaces, and let $T : V \rightarrow W$ be a linear transformation. Given a subspace $U$ of $V,$ let $T(U)$ denote the set of all images of the form $T(\mathbf{x}),$ where $\mathbf{x}$ is in $U .$ Show that $T(U)$ is a subspace of $W .$

Michael J.

Numerade Educator

Problem 36

Given $T : V \rightarrow W$ as in Exercise $35,$ and given a subspace $Z$ of $W,$ let $U$ be the set of all $\mathbf{x}$ in $V$ such that $T(\mathbf{x})$ is in $Z$ Show that $U$ is a subspace of $V .$

Doruk I.

Numerade Educator

Problem 37

[MJ Determine whether $w$ is in the column space of $A,$ the null space of $A,$ or both, where
$$
\mathbf{w}=\left[\begin{array}{r}{1} \\ {1} \\ {-1} \\ {-3}\end{array}\right], \quad A=\left[\begin{array}{rrrr}{7} & {6} & {-4} & {1} \\ {-5} & {-1} & {0} & {-2} \\ {9} & {-11} & {7} & {-3} \\ {19} & {-9} & {7} & {1}\end{array}\right]
$$

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Problem 38

[M] Determine whether $\mathbf{w}$ is in the column space of $A,$ the null space of $A,$ or both, where
$$
\mathbf{w}=\left[\begin{array}{l}{1} \\ {2} \\ {1} \\ {0}\end{array}\right], \quad A=\left[\begin{array}{rrrr}{-8} & {5} & {-2} & {0} \\ {-5} & {2} & {1} & {-2} \\ {10} & {-8} & {6} & {-3} \\ {3} & {-2} & {1} & {0}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 39

$[\mathbf{M}]$ Let $\mathbf{a}_{1}, \ldots, \mathbf{a}_{5}$ denote the columns of the matrix $A,$ where
$$
A=\left[\begin{array}{rrrrr}{5} & {1} & {2} & {2} & {0} \\ {3} & {3} & {2} & {-1} & {-12} \\ {8} & {4} & {4} & {-5} & {12} \\ {2} & {1} & {1} & {0} & {-2}\end{array}\right], \quad B=\left[\begin{array}{ccc}{\mathbf{a}_{1}} & {\mathbf{a}_{2}} & {\mathbf{a}_{4}}\end{array}\right]
$$
a. Explain why $a_{3}$ and $a_{5}$ are in the column space of $B$
b. Find a set of vectors that spans Nul $A .$
c. Let $T : \mathbb{R}^{5} \rightarrow \mathbb{R}^{4}$ be defined by $T(\mathbf{x})=A \mathbf{x} .$ Explain why $T$ is neither one-to-one nor onto.

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Problem 40

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ and $K=\operatorname{Span}\left\{\mathbf{v}_{3}, \mathbf{v}_{4}\right\},$ where
$$
\mathbf{v}_{1}=\left[\begin{array}{l}{5} \\ {3} \\ {8}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{l}{1} \\ {3} \\ {4}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{2} \\ {-1} \\ {5}\end{array}\right], \mathbf{v}_{4}=\left[\begin{array}{r}{0} \\ {-12} \\ {-28}\end{array}\right]
$$
Then $H$ and $K$ are subspaces of $\mathbb{R}^{3} .$ In fact, $H$ and $K$ are planes in $\mathbb{R}^{3}$ through the origin, and they intersect in a line through $0 .$ Find a nonzero vector $\mathbf{w}$ that generates that line. [Hine. [Hint: $\mathbf{w}$ can be written as $c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}$

Doruk I.

Numerade Educator