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Linear Algebra

Seymour Lipschutz, Marc Lipson

Chapter 5

Linear Mappings - all with Video Answers

Educators

WM

Chapter Questions

02:12

Problem 1

State whether each diagram in Fig. $5-3$ defines a mapping from $A=\{a, b, c\}$ into $B=\{x, y, z\}.$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:12

Problem 2

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be defined by Fig. $5-4.$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:10

Problem 3

Consider the mapping $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=\left(y z, x^{2}\right) .$ Find
(a) $F(2,3,4)$
(b) $F(5,-2,7)$
(c) $F^{-1}(0,0)$, that is, all $v \in \mathbf{R}^{3}$ such that $F(v)=0$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:14

Problem 4

Consider the mapping $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(3 y, 2 x) .$ Let $S$ be the unit circle in $\mathbf{R}^{2}$ that is, the solution set of $x^{2}+y^{2}=1$
(a) Describe $F(S)$. (b) Find $F^{-1}(S)$

Anthony Ramos
Anthony Ramos
Numerade Educator
08:03

Problem 5

Let the mappings $f: A \rightarrow B, g: B \rightarrow C, h: C \rightarrow D$ be defined by Fig. 5-5. Determine whether or not each function is (a) one-to-one; (b) onto;
(c) invertible (i.e., has an inverse).

Anthony Ramos
Anthony Ramos
Numerade Educator
14:32

Problem 6

Suppose $f: A \rightarrow B$ and $g: B \rightarrow C .$ Hence, $(g \circ f): A \rightarrow C$ exists. Prove
(a) If $f$ and $g$ are one-to-one, then $g \circ f$ is one-to-one.
(b) If $f$ and $g$ are onto mappings, then $g \circ f$ is an onto mapping.
(c) If $g \circ f$ is one-to-one, then $f$ is one-to-one.
(d) If $g \circ f$ is an onto mapping, then $g$ is an onto mapping.

Anthony Ramos
Anthony Ramos
Numerade Educator
07:07

Problem 7

Prove that $f: A \rightarrow B$ has an inverse if and only if $f$ is one-to-one and onto.

WM
William Mead
Numerade Educator
02:12

Problem 8

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=2 x-3 .$ Now $f$ is one-to-one and onto; hence, $f$ has an inverse mapping $f^{-1}$. Find a formula for $f^{-1}.$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:54

Problem 9

Suppose the mapping $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ is defined by $F(x, y)=(x+y, x) .$ Show that $F$ is linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:57

Problem 10

Suppose $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ is defined by $F(x, y, z)=(x+y+z, 2 x-3 y+4 z) .$ Show that $F$ is linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:47

Problem 11

Show that the following mappings are not linear:
(a) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(x y, x)$
(b) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}$ defined by $F(x, y)=(x+3,2 y, x+y)$
(c) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=(|x|, y+z)$
(a) Let $v=(1,2)$ and $w=(3,4) ;$ then $v+w=(4,6) .$ Also,
\[
F(v)=(1(2), 1)=(2,1) \quad \text { and } \quad F(w)=(3(4), 3)=(12,3)
\]
Hence,
\[
F(v+w)=(4(6), 4)=(24,6) \neq F(v)+F(w)
\]
(b) Because $F(0,0)=(3,0,0) \neq(0,0,0), F$ cannot be linear.
(c) Let $v=(1,2,3)$ and $k=-3 .$ Then $k v=(-3,-6,-9) .$ We have
\[
F(v)=(1,5) \text { and } k F(v)=-3(1,5)=(-3,-15)
\]
Thus,
\[
F(k v)=F(-3,-6,-9)=(3,-15) \neq k F(v)
\]
Accordingly, $F$ is not linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:47

Problem 11

Show that the following mappings are not linear:
(a) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(x y, x)$
(b) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}$ defined by $F(x, y)=(x+3,2 y, x+y)$
(c) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=(|x|, y+z)$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:03

Problem 12

Let $V$ be the vector space of $n$ -square real matrices. Let $M$ be an arbitrary but fixed matrix in $V$ Let $F: V \rightarrow V$ be defined by $F(A)=A M+M A$, where $A$ is any matrix in $V$. Show that $F$ is linear.
For any matrices $A$ and $B$ in $V$ and any scalar $k$, we have
\[
\begin{aligned}
F(A+B) &=(A+B) M+M(A+B)=A M+B M+M A+M B \\
&=(A M+M A)=(B M+M B)=F(A)+F(B)
\end{aligned}
\]
and
\[
F(k A)=(k A) M+M(k A)=k(A M)+k(M A)=k(A M+M A)=k F(A)
\]
Thus, $F$ is linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
05:03

Problem 12

Let $V$ be the vector space of $n$ -square real matrices. Let $M$ be an arbitrary but fixed matrix in $V$ Let $F: V \rightarrow V$ be defined by $F(A)=A M+M A$, where $A$ is any matrix in $V .$ Show that $F$ is linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
14:08

Problem 13

Prove Theorem 5.2: Let $V$ and $U$ be vector spaces over a field $K$. Let $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ be a basis of $V$ and let $u_{1}, u_{2}, \ldots, u_{n}$ be any vectors in $U .$ Then there exists a unique linear mapping $F: V \rightarrow U$ such that $F\left(v_{1}\right)=u_{1}, F\left(v_{2}\right)=u_{2}, \ldots, F\left(v_{n}\right)=u_{n}$

Anthony Ramos
Anthony Ramos
Numerade Educator
14:09

Problem 13

Prove Theorem 5.2: Let $V$ and $U$ be vector spaces over a field $K .$ Let $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ be a basis of $V$ and let $u_{1}, u_{2}, \ldots, u_{n}$ be any vectors in $U .$ Then there exists a unique linear mapping $F: V \rightarrow U$ such that $F\left(v_{1}\right)=u_{1}, F\left(v_{2}\right)=u_{2}, \ldots, F\left(v_{n}\right)=u_{n}$
There are three steps to the proof of the theorem: (1) Define the mapping $F: V \rightarrow U$ such that $F\left(v_{i}\right)=u_{i}, i=1, \ldots, n \cdot(2)$ Show that $F$ is linear. (3) Show that $F$ is unique.

Step 1. Let $v \in V .$ Because $\left\{v_{1}, \ldots, v_{n}\right\}$ is a basis of $V,$ there exist unique scalars $a_{1}, \ldots, a_{n} \in K$ for which $v=a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{n} v_{n} .$ We define $F: V \rightarrow U$ by
\[
F(v)=a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} u_{n}
\]
(Because the $a_{i}$ are unique, the mapping $F$ is well defined.) Now, for $i=1, \ldots, n$
\[
v_{i}=0 v_{1}+\cdots+1 v_{i}+\cdots+0 v_{n}
\]
Hence,
\[
F\left(v_{i}\right)=0 u_{1}+\cdots+1 u_{i}+\cdots+0 u_{n}=u_{i}
\]
Thus, the first step of the proof is complete.
Step
2. Suppose $v=a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{n} v_{n}$ and $w=b_{1} v_{1}+b_{2} v_{2}+\cdots+b_{n} v_{n} .$ Then
\[
v+w=\left(a_{1}+b_{1}\right) v_{1}+\left(a_{2}+b_{2}\right) v_{2}+\cdots+\left(a_{n}+b_{n}\right) v_{n}
\]
and, for any $k \in K, k v=k a_{1} v_{1}+k a_{2} v_{2}+\cdots+k a_{n} v_{n} .$ By definition of the mapping $F$
\[
F(v)=a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} v_{n} \quad \text { and } \quad F(w)=b_{1} u_{1}+b_{2} u_{2}+\cdots+b_{n} u_{n}
\]
Hence,
\[
\begin{aligned}
F(v+w) &=\left(a_{1}+b_{1}\right) u_{1}+\left(a_{2}+b_{2}\right) u_{2}+\cdots+\left(a_{n}+b_{n}\right) u_{n} \\
&=\left(a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} u_{n}\right)+\left(b_{1} u_{1}+b_{2} u_{2}+\cdots+b_{n} u_{n}\right) \\
&=F(v)+F(w)
\end{aligned}
\]
and
\[
F(k v)=k\left(a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} u_{n}\right)=k F(v)
\]
Thus, $F$ is linear.
Step
3. Suppose $G: V \rightarrow U$ is linear and $G\left(v_{1}\right)=u_{i}, i=1, \ldots, n$. Let
\[
v=a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{n} v_{n}
\]
Then
\[
\begin{aligned}
G(v) &=G\left(a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{n} v_{n}\right)=a_{1} G\left(v_{1}\right)+a_{2} G\left(v_{2}\right)+\cdots+a_{n} G\left(v_{n}\right) \\
&=a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} u_{n}=F(v)
\end{aligned}
\]
Because $G(v)=F(v)$ for every $v \in V, G=F .$ Thus, $F$ is unique and the theorem is proved.

Anthony Ramos
Anthony Ramos
Numerade Educator
06:58

Problem 14

Let $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be the linear mapping for which $F(1,2)=(2,3)$ and $F(0,1)=(1,4) .[$ Note that $\left.\{(1,2),(0,1)\} \text { is a basis of } \mathbf{R}^{2}, \text { so such a linear map } F \text { exists and is unique by Theorem } 5.2 .\right]$ Find a formula for $F ;$ that is, find $F(a, b).$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:41

Problem 15

Suppose a linear mapping $F: V \rightarrow U$ is one-to-one and onto. Show that the inverse mapping $F^{-1}: U \rightarrow V$ is also linear.

Anthony Ramos
Anthony Ramos
Numerade Educator
18:41

Problem 16

Let $F: \mathbf{R}^{4} \rightarrow \mathbf{R}^{3}$ be the linear mapping defined by
\[
F(x, y, z, t)=(x-y+z+t, \quad x+2 z-t, \quad x+y+3 z-3 t)
\]
Find a basis and the dimension of (a) the image of $F,$ (b) the kernel of $F.$

Anthony Ramos
Anthony Ramos
Numerade Educator
11:01

Problem 17

Let $G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ be the linear mapping defined by
\[
G(x, y, z)=(x+2 y-z, \quad y+z, \quad x+y-2 z)
\]
Find a basis and the dimension of (a) the image of $G,(\mathrm{b})$ the kernel of $G$

Anthony Ramos
Anthony Ramos
Numerade Educator
09:30

Problem 18

Consider the matrix mapping $A: \mathbf{R}^{4} \rightarrow \mathbf{R}^{3},$ where $A=\left[\begin{array}{rrrr}1 & 2 & 3 & 1 \\ 1 & 3 & 5 & -2 \\ 3 & 8 & 13 & -3\end{array}\right] .$ Find a basis and the dimension of (a) the image of $A,(\mathrm{b})$ the kernel of $A.$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:28

Problem 19

Find a linear map $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{4}$ whose image is spanned by (1,2,0,-4) and (2,0,-1,-3).

Anthony Ramos
Anthony Ramos
Numerade Educator
07:44

Problem 20

Suppose $f: V \rightarrow U$ is linear with kernel $W$, and that $f(v)=u$. Show that the "coset" $v+W=\{v+w: w \in W\}$ is the preimage of $u ;$ that is, $f^{-1}(u)=v+W.$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:53

Problem 21

Suppose $F: V \rightarrow U$ and $G: U \rightarrow W$ are linear. Prove
(a) $\operatorname{rank}(G \circ F) \leq \operatorname{rank}(G)$
(b) $\operatorname{rank}(G \circ F) \leq \operatorname{rank}(F)$

Anthony Ramos
Anthony Ramos
Numerade Educator
07:30

Problem 22

Prove Theorem 5.3: Let $F: V \rightarrow U$ be linear. Then,
(a) $\operatorname{Im} F$ is a subspace of $U$
(b) Ker $F$ is a subspace of $V$

Anthony Ramos
Anthony Ramos
Numerade Educator
22:57

Problem 23

Prove Theorem 5.6: Suppose $V$ has finite dimension and $F: V \rightarrow U$ is linear. Then
\[
\operatorname{dim} V=\operatorname{dim}(\operatorname{Ker} F)+\operatorname{dim}(\operatorname{Im} F)=\operatorname{nullity}(F)+\operatorname{rank}(F)
\]

Anthony Ramos
Anthony Ramos
Numerade Educator
05:37

Problem 24

Determine whether or not each of the following linear maps is nonsingular. If not, find a nonzero vector $v$ whose image is 0
(a) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(x-y, x-2 y)$
(b) $G: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $G(x, y)=(2 x-4 y, 3 x-6 y)$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:41

Problem 25

The linear $\operatorname{map} F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(x-y, x-2 y)$ is nonsingular by the previous Problem $5.24 .$ Find a formula for $F^{-1}$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:10

Problem 26

Let $G: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}$ be defined by $G(x, y)=(x+y, x-2 y, 3 x+y)$
(a) Show that $G$ is nonsingular.
(b) Find a formula for $G^{-1}$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:24

Problem 27

Suppose that $F: V \rightarrow U$ is linear and that $V$ is of finite dimension. Show that $V$ and the image of $F$ have the same dimension if and only if $F$ is nonsingular. Determine all nonsingular linear mappings $T: \mathbf{R}^{4} \rightarrow \mathbf{R}^{3}.$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:56

Problem 28

Prove Theorem 5.7: Let $F: V \rightarrow U$ be a nonsingular linear mapping. Then the image of any linearly independent set is linearly independent.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:07

Problem 29

Prove Theorem 5.9: Suppose $V$ has finite dimension and $\operatorname{dim} V=\operatorname{dim} U .$ Suppose $F: V \rightarrow U$ is linear. Then $F$ is an isomorphism if and only if $F$ is nonsingular.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:41

Problem 30

Define $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ and $G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ by $F(x, y, z)=(2 x, y+z)$ and $G(x, y, z)=(x-z, y)$
Find formulas defining the maps:
(b) $3 F$
(a) $F+G$
(c) $2 F-5 G$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:16

Problem 31

Let $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ and $G: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be defined by $F(x, y, z)=(2 x, y+z)$ and $G(x, y)=(y, x)$
Derive formulas defining the mappings: (a) $G \circ F,$ (b) $F \circ G$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:40

Problem 32

Prove: (a) The zero mapping $0,$ defined by $\mathbf{0}(v)=0 \in U$ for every $v \in V,$ is the zero element of $\operatorname{Hom}(V, U) .$ (b) The negative of $F \in \operatorname{Hom}(V, U)$ is the mapping $(-1) F,$ that is, $-F=(-1) F$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:51

Problem 33

Suppose $F_{1}, F_{2}, \ldots, F_{n}$ are linear maps from $V$ into $U .$ Show that, for any scalars $a_{1}, a_{2}, \ldots, a_{n}$ and for any $v \in V$
\[
\left(a_{1} F_{1}+a_{2} F_{2}+\cdots+a_{n} F_{n}\right)(v)=a_{1} F_{1}(v)+a_{2} F_{2}(v)+\cdots+a_{n} F_{n}(v)
\]

Anthony Ramos
Anthony Ramos
Numerade Educator
10:07

Problem 34

Consider linear mappings $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}, \quad G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}, H: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by
$F(x, y, z)=(x+y+z, x+y), \quad G(x, y, z)=(2 x+z, x+y), \quad H(x, y, z)=(2 y, x)$
Show that $\left.F, G, H \text { are linearly independent [as elements of } \operatorname{Hom}\left(\mathbf{R}^{3}, \mathbf{R}^{2}\right)\right]$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:09

Problem 35

Let $k$ be a nonzero scalar. Show that a linear map $T$ is singular if and only if $k T$ is singular. Hence, $T$ is singular if and only if $-T$ is singular.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:48

Problem 36

Find the dimension $d$ of:
(a) $\operatorname{Hom}\left(\mathbf{R}^{3}, \mathbf{R}^{4}\right)$
(b) $\operatorname{Hom}\left(\mathbf{R}^{5}, \mathbf{R}^{3}\right)$
(c) $\operatorname{Hom}\left(\mathbf{P}_{3}(t), \mathbf{R}^{2}\right), \quad(d) \operatorname{Hom}\left(\mathbf{M}_{2,3}, \mathbf{R}^{4}\right)$
Use $\operatorname{dim}[\operatorname{Hom}(V, U)]=m n,$ where $\operatorname{dim} V=m$ and $\operatorname{dim} U=n.$

Anthony Ramos
Anthony Ramos
Numerade Educator
18:33

Problem 37

Prove Theorem $5.11 .$ Suppose $\operatorname{dim} V=m$ and $\operatorname{dim} U=n .$ Then $\operatorname{dim}[\operatorname{Hom}(V, U)]=m n.$

Anthony Ramos
Anthony Ramos
Numerade Educator
08:19

Problem 38

Prove Theorem 5.12:
(i) $G \circ\left(F+F^{\prime}\right)=G \circ F+G \circ F^{\prime}$
(ii) $\left(G+G^{\prime}\right) \circ F=G \circ F+G^{\prime} \circ F$
(iii) $k(G \circ F)=(k G) \circ F=G \circ(k F)$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:10

Problem 39

Let $F$ and $G$ be the linear operators on $\mathbf{R}^{2}$ defined by $F(x, y)=(y, x)$ and $G(x, y)=(0, x) .$ Find formulas defining the following operators:
(a) $F+G$
(b) $2 F-3 G$
(c) $F G$
(d) $G F$
$(\mathrm{e}) F^{2}$
(f) $G^{2}$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:46

Problem 40

Consider the linear operator $T$ on $\mathbf{R}^{3}$ defined by $T(x, y, z)=(2 x, 4 x-y, 2 x+3 y-z)$
(a) Show that $T$ is invertible. Find formulas for (b) $T^{-1}$,
$(\mathrm{c}) T^{2},(d) T^{-2}$

Anthony Ramos
Anthony Ramos
Numerade Educator
10:43

Problem 41

Let $V$ be of finite dimension and let $T$ be a linear operator on $V$ for which $T R=I$, for some operator $R$ on $V$. (We call $R$ a right inverse of $T$.)
(a) Show that $T$ is invertible.
(b) Show that $R=T^{-1}$
(c) Give an example showing that the above need not hold if $V$ is of infinite dimension.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:00

Problem 42

Let $F$ and $G$ be linear operators on $\mathbf{R}^{2}$ defined by $F(x, y)=(0, x)$ and $G(x, y)=(x, 0) .$ Show that (a) $G F=\mathbf{0},$ the zero mapping, but $F G \neq \mathbf{0} .$ (b) $G^{2}=G.$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:00

Problem 43

Find the dimension of $(\mathrm{a}) A\left(\mathbf{R}^{4}\right),(\mathrm{b}) A\left(\mathbf{P}_{2}(t)\right),(\mathrm{c}) A\left(\mathbf{M}_{2,3}\right)$

Anthony Ramos
Anthony Ramos
Numerade Educator
10:33

Problem 44

Let $E$ be a linear operator on $V$ for which $E^{2}=E .$ (Such an operator is called a projection.) Let $U$ be the image of $E,$ and let $W$ be the kernel. Prove
(a) If $u \in U,$ then $E(u)=u$ (i.e., $E$ is the identity mapping on $U$ ).
(b) If $E \neq I,$ then $E$ is singular-that is, $E(v)=0$ for some $v \neq 0$
(c) $\quad V=U \oplus W$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:12

Problem 45

Determine the number of different mappings from $(a)\{1,2\}$ into $\{1,2,3\},(b)\{1,2, \ldots, r\}$ into $\{1,2, \ldots, s\}$

Srilakshmi E K
Srilakshmi E K
Numerade Educator
04:47

Problem 46

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x^{2}+3 x+1$ and $g(x)=2 x-3 .$ Find formulas defining the composition mappings:
(a) $f \circ g ;$ (b) $g \circ f ;$ (c) $g \circ g ;(d) f \circ f$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:40

Problem 47

For each mappings $f: \mathbf{R} \rightarrow \mathbf{R}$ find a formula for its inverse:
(a) $f(x)=3 x-7,$ (b) $f(x)=x^{3}+2$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:22

Problem 48

For any mapping $f: A \rightarrow B,$ show that $1_{B} \circ f=f=f \circ \mathbf{1}_{A}$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:47

Problem 49

Show that the following mappings are linear:
(a) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=(x+2 y-3 z, 4 x-5 y+6 z)$
(b) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(a x+b y, c x+d y),$ where $a, b, c, d$ belong to $\mathbf{R}$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:47

Problem 50

Show that the following mappings are not linear:
(a) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=\left(x^{2}, y^{2}\right)$
(b) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=(x+1, y+z)$
(c) $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y)=(x y, y)$
(d) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $F(x, y, z)=(|x|, y+z)$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:58

Problem 51

Find $F(a, b),$ where the linear map $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ is defined by $F(1,2)=(3,-1)$ and $F(0,1)=(2,1).$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:08

Problem 52

Find a $2 \times 2$ matrix $A$ that maps
(a) $\quad(1,3)^{T}$ and $(1,4)^{T}$ into $(-2,5)^{T}$ and $(3,-1)^{T},$ respectively
(b) $(2,-4)^{T}$ and $(-1,2)^{T}$ into $(1,1)^{T}$ and $(1,3)^{T},$ respectively.

Ashley Boni
Ashley Boni
Numerade Educator
02:05

Problem 53

Find a $2 \times 2$ singular matrix $B$ that maps $(1,1)^{T}$ into $(1,3)^{T}.$

Jacob Denson
Jacob Denson
Numerade Educator
05:03

Problem 54

Let $V$ be the vector space of real $n$ -square matrices, and let $M$ be a fixed nonzero matrix in $V$. Show that the first two of the following mappings $T: V \rightarrow V$ are linear, but the third is not:
(a) $T(A)=M A,$ (b) $T(A)=A M+M A$, (c) $T(A)=M+A$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:44

Problem 55

Give an example of a nonlinear map $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ such that $F^{-1}(0)=\{0\}$ but $F$ is not one-to-one.

Kayla Robinson
Kayla Robinson
Numerade Educator
06:14

Problem 56

$\operatorname{Let} F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be defined by $F(x, y)=(3 x+5 y, 2 x+3 y)$, and let $S$ be the unit circle in $\mathbf{R}^{2}$. ( $S$ consists of all points satisfying $x^{2}+y^{2}=1 .$ ) Find (a) the image $F(S),$ (b) the preimage $F^{-1}(S)$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:34

Problem 57

Consider the linear map $G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ defined by $G(x, y, z)=(x+y+z, y-2 z, y-3 z)$ and the unit sphere $S_{2}$ in $\mathbf{R}^{3},$ which consists of the points satisfying $x^{2}+y^{2}+z^{2}=1 .$ Find $(\mathrm{a}) G\left(S_{2}\right),$ (b) $G^{-1}\left(S_{2}\right)$

A M
A M
Numerade Educator
08:31

Problem 58

Let $H$ be the plane $x+2 y-3 z=4$ in $\mathbf{R}^{3}$ and let $G$ be the linear map in Problem 5.57 . Find
(a) $G(H)$
(b) $G^{-1}(H)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator

Problem 59

Let $W$ be a subspace of $V$. The inclusion map, denoted by $i: W \hookrightarrow V$, is defined by $i(w)=w$ for every $w \in W .$ Show that the inclusion map is linear.

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05:41

Problem 60

Suppose $F: V \rightarrow U$ is linear. Show that $F(-v)=-F(v).$

Anthony Ramos
Anthony Ramos
Numerade Educator
18:41

Problem 61

For each linear map $F$ find a basis and the dimension of the kernel and the image of $F$
(a) $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ defined by $F(x, y, z)=(x+2 y-3 z, 2 x+5 y-4 z, x+4 y+z)$
(b) $F: \mathbf{R}^{4} \rightarrow \mathbf{R}^{3}$ defined by $F(x, y, z, t)=(x+2 y+3 z+2 t, 2 x+4 y+7 z+5 t, x+2 y+6 z+5 t)$

Anthony Ramos
Anthony Ramos
Numerade Educator
11:01

Problem 62

For each linear map $G,$ find a basis and the dimension of the kernel and the image of $G:$
(a) $\quad G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $G(x, y, z)=(x+y+z, 2 x+2 y+2 z)$
(b) $\quad G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ defined by $G(x, y, z)=(x+y, y+z)$
(c) $\quad G: \mathbf{R}^{5} \rightarrow \mathbf{R}^{3}$ defined by
\[
G(x, y, z, s, t)=(x+2 y+2 z+s+t, \quad x+2 y+3 z+2 s-t, \quad 3 x+6 y+8 z+5 s-t)
\]

Anthony Ramos
Anthony Ramos
Numerade Educator
09:30

Problem 63

Each of the following matrices determines a linear map from $\mathbf{R}^{4}$ into $\mathbf{R}^{3}$ :
(a) $A=\left[\begin{array}{rrrr}1 & 2 & 0 & 1 \\ 2 & -1 & 2 & -1 \\ 1 & -3 & 2 & -2\end{array}\right],$ (b) $B=\left[\begin{array}{rrrr}1 & 0 & 2 & -1 \\ 2 & 3 & -1 & 1 \\ -2 & 0 & -5 & 3\end{array}\right]$
Find a basis as well as the dimension of the kernel and the image of each linear map.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:15

Problem 64

Find a linear mapping $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ whose image is spanned by (1,2,3) and (4,5,6).

Nick Johnson
Nick Johnson
Numerade Educator
18:41

Problem 65

Find a linear mapping $G: \mathbf{R}^{4} \rightarrow \mathbf{R}^{3}$ whose kernel is spanned by (1,2,3,4) and (0,1,1,1)

Anthony Ramos
Anthony Ramos
Numerade Educator
09:02

Problem 66

Let $V=\mathbf{P}_{10}(t),$ the vector space of polynomials of degree $\leq 10 .$ Consider the linear map $\mathbf{D}^{4}: V \rightarrow V$, where $\mathbf{D}^{4}$ denotes the fourth derivative $d^{4}(f) / d t^{4} .$ Find a basis and the dimension of
(a) the image of $\mathbf{D}^{4} ;$ (b) the kernel of $\mathbf{D}^{4}$

Will Erickson
Will Erickson
Numerade Educator
07:30

Problem 67

Suppose $F: V \rightarrow U$ is linear. Show that $(a)$ the image of any subspace of $V$ is a subspace of $U$ (b) the preimage of any subspace of $U$ is a subspace of $V.$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:41

Problem 68

Show that if $F: V \rightarrow U$ is onto, then $\operatorname{dim} U \leq \operatorname{dim} V$. Determine all linear maps $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{4}$ that are onto.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:20

Problem 69

Consider the zero mapping $\mathbf{0}: V \rightarrow U$ defined by $\mathbf{0}(v)=0, \forall v \in V .$ Find the kernel and the image of $\mathbf{0}.$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:16

Problem 70

Let $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ and $G: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}$ be defined by $F(x, y, z)=(y, x+z)$ and $G(x, y, z)=(2 z, x-y) .$ Find
formulas defining the mappings $F+G$ and $3 F-2 G$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:41

Problem 71

Let $H: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ be defined by $H(x, y)=(y, 2 x) .$ Using the maps $F$ and $G$ in Problem 5.70 , find formulas
(c) $H \circ(F+G)$ and $H \circ F+H \circ G$ defining the mappings: (a) $H \circ F$ and $H \circ G,$ (b) $F \circ H$ and $G \circ H,$

Anthony Ramos
Anthony Ramos
Numerade Educator
10:07

Problem 72

Show that the following mappings $F, G, H$ are linearly independent:
(a) $F, G, H \in \operatorname{Hom}\left(\mathbf{R}^{2}, \mathbf{R}^{2}\right)$ defined by $F(x, y)=(x, 2 y), \quad G(x, y)=(y, x+y), \quad H(x, y)=(0, x)$
(b) $\quad F, G, H \in \operatorname{Hom}\left(\mathbf{R}^{3}, \mathbf{R}\right)$ defined by $F(x, y, z)=x+y+z, \quad G(x, y, z)=y+z, \quad H(x, y, z)=x-z$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:56

Problem 73

For $F, G \in \operatorname{Hom}(V, U),$ show that $\operatorname{rank}(F+G) \leq \operatorname{rank}(F)+\operatorname{rank}(G) .$ (Here $V$ has finite dimension.)

Nick Johnson
Nick Johnson
Numerade Educator
04:07

Problem 74

Let $F: V \rightarrow U$ and $G: U \rightarrow V$ be linear. Show that if $F$ and $G$ are nonsingular, then $G \circ F$ is nonsingular Give an example where $G \circ F$ is nonsingular but $G$ is not. [Hint: Let $\operatorname{dim} V<\operatorname{dim} U .$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:00

Problem 75

Find the dimension $d$ of $(a) \operatorname{Hom}\left(\mathbf{R}^{2}, \mathbf{R}^{8}\right),(b) \operatorname{Hom}\left(\mathbf{P}_{4}(t), \mathbf{R}^{3}\right),(\mathrm{c}) \operatorname{Hom}\left(\mathbf{M}_{2,4}, \mathbf{P}_{2}(t)\right)$

Anthony Ramos
Anthony Ramos
Numerade Educator
10:33

Problem 76

5.84. Suppose $F: V \rightarrow U$ is linear and $k$ is a nonzero scalar. Prove that the maps $F$ and $k F$ have the same kernel and the same image.
5.85. Suppose $F$ and $G$ are linear operators on $V$ and that $F$ is nonsingular. Assume that $V$ has finite dimension. Show that $\operatorname{rank}(F G)=\operatorname{rank}(G F)=\operatorname{rank}(G)$
5.86. Suppose $V$ has finite dimension. Suppose $T$ is a linear operator on $V$ such that $\operatorname{rank}\left(T^{2}\right)=\operatorname{rank}(T) .$ Show that $\operatorname{Ker} T \cap \operatorname{Im} T=\{0\}$
5.87. Suppose $V=U \oplus W$. Let $E_{1}$ and $E_{2}$ be the linear operators on $V$ defined by $E_{1}(v)=u, E_{2}(v)=w,$ where $v=u+w, u \in U, w \in W .$ Show that
(a) $E_{1}^{2}=E_{1}$ and $E_{2}^{2}=E_{2}$ (i.e., that $E_{1}$ and $E_{2}$ are projections)
(b) $E_{1}+E_{2}=I,$ the identity mapping;
(c) $E_{1} E_{2}=\mathbf{0}$ and $E_{2} E_{1}=\mathbf{0}$
5.88. Let $E_{1}$ and $E_{2}$ be linear operators on $V$ satisfying parts (a), (b), (c) of Problem 5.88. Prove
\[
V=\operatorname{Im} E_{1} \oplus \operatorname{Im} E_{2}
\]
5.89. Let $v$ and $w$ be elements of a real vector space $V$. The line segment $L$ from $v$ to $v+w$ is defined to be the set of vectors $v+t w$ for $0 \leq t \leq 1$. (See Fig. $5.6 .$
(a) Show that the line segment $L$ between vectors $v$ and $u$ consists of the points:
(i) $(1-t) v+t u$ for $0 \leq t \leq 1$
(ii) $t_{1} v+t_{2} u$ for $t_{1}+t_{2}=1, t_{1} \geq 0, t_{2} \geq 0$
(b) Let $F: V \rightarrow U$ be linear. Show that the image $F(L)$ of a line segment $L$ in $V$ is a line segment in $U$

Anthony Ramos
Anthony Ramos
Numerade Educator
18:33

Problem 77

When can dim [Hom $(V, U)]=\operatorname{dim} V ?$

Anthony Ramos
Anthony Ramos
Numerade Educator
06:10

Problem 78

Let $F$ and $G$ be the linear operators on $\mathbf{R}^{2}$ defined by $F(x, y)=(x+y, 0)$ and $G(x, y)=(-y, x) .$ Find formulas defining the linear operators:
(a) $F+G,$ (b) $5 F-3 G,$ (c) $F G,(d) G F,(e) F^{2},(f) G^{2}$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:46

Problem 79

Show that each linear operator $T$ on $\mathbf{R}^{2}$ is nonsingular and find a formula for $T^{-1}$, where
(a) $T(x, y)=(x+2 y, 2 x+3 y),$ (b) $T(x, y)=(2 x-3 y, 3 x-4 y)$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:46

Problem 80

Show that each of the following linear operators $T$ on $\mathbf{R}^{3}$ is nonsingular and find a formula for $T^{-1}$, where
(a) $T(x, y, z)=(x-3 y-2 z, y-4 z, z)$
(b) $T(x, y, z)=(x+z, x-y, y)$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:00

Problem 81

Find the dimension of $A(V),$ where (a) $V=\mathbf{R}^{7},$ (b) $V=\mathbf{P}_{5}(t),$ (c) $V=\mathbf{M}_{3,4}$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:51

Problem 82

Which of the following integers can be the dimension of an algebra $A(V)$ of linear maps:
$5,9,12,25,28,36,45,64,88,100 ?$

Supratim Roy
Supratim Roy
Numerade Educator
05:46

Problem 83

Let $T$ be the linear operator on $\mathbf{R}^{2}$ defined by $T(x, y)=(x+2 y, 3 x+4 y) .$ Find a formula for $f(T),$ where
(a) $f(t)=t^{2}+2 t-3$
(b) $f(t)=t^{2}-5 t-2$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:51

Problem 84

Suppose $F: V \rightarrow U$ is linear and $k$ is a nonzero scalar. Prove that the maps $F$ and $k F$ have the same kernel and the same image.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:56

Problem 85

Suppose $F$ and $G$ are linear operators on $V$ and that $F$ is nonsingular. Assume that $V$ has finite dimension. Show that $\operatorname{rank}(F G)=\operatorname{rank}(G F)=\operatorname{rank}(G)$

Nick Johnson
Nick Johnson
Numerade Educator
01:56

Problem 86

Suppose $V$ has finite dimension. Suppose $T$ is a linear operator on $V$ such that $\operatorname{rank}\left(T^{2}\right)=\operatorname{rank}(T) .$ Show that $\operatorname{Ker} T \cap \operatorname{Im} T=\{0\}$

Nick Johnson
Nick Johnson
Numerade Educator
10:33

Problem 87

Suppose $V=U \oplus W$. Let $E_{1}$ and $E_{2}$ be the linear operators on $V$ defined by $E_{1}(v)=u, E_{2}(v)=w,$ where $v=u+w, u \in U, w \in W .$ Show that
(a) $E_{1}^{2}=E_{1}$ and $E_{2}^{2}=E_{2}$ (i.e., that $E_{1}$ and $E_{2}$ are projections)
(b) $E_{1}+E_{2}=I,$ the identity mapping;
(c) $E_{1} E_{2}=\mathbf{0}$ and $E_{2} E_{1}=\mathbf{0}$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:00

Problem 88

Let $E_{1}$ and $E_{2}$ be linear operators on $V$ satisfying parts (a), (b), (c) of Problem 5.88. Prove
\[
V=\operatorname{Im} E_{1} \oplus \operatorname{Im} E_{2}
\]

Hoan Nguyen
Hoan Nguyen
Numerade Educator
03:23

Problem 89

Let $v$ and $w$ be elements of a real vector space $V$. The line segment $L$ from $v$ to $v+w$ is defined to be the set of vectors $v+t w$ for $0 \leq t \leq 1$. (See Fig. $5.6 .$
(a) Show that the line segment $L$ between vectors $v$ and $u$ consists of the points:
(i) $(1-t) v+t u$ for $0 \leq t \leq 1$
(ii) $t_{1} v+t_{2} u$ for $t_{1}+t_{2}=1, t_{1} \geq 0, t_{2} \geq 0$
(b) Let $F: V \rightarrow U$ be linear. Show that the image $F(L)$ of a line segment $L$ in $V$ is a line segment in $U$

Uma Kumari
Uma Kumari
Numerade Educator
07:30

Problem 90

5.90. Let $F: V \rightarrow U$ be linear and let $W$ be a subspace of $V$. The restriction of $F$ to $W$ is the $\operatorname{map} F | W: W \rightarrow U$ defined by $F | W(v)=F(v)$ for every $v$ in $W$. Prove the following:
(a) $F | W$ is linear;
(b) $\operatorname{Ker}(F | W)=(\operatorname{Ker} F) \cap W$
(c) $\operatorname{Im}(F | W)=F(W)$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:07

Problem 91

A subset $X$ of a vector space $V$ is said to be convex if the line segment $L$ between any two points (vectors) $P, Q \in X$ is contained in $X$. (a) Show that the intersection of convex sets is convex; (b) suppose $F: V \rightarrow U$ is linear and $X$ is convex. Show that $F(X)$ is convex.

Lucía Guerrero
Lucía Guerrero
Numerade Educator