Chapter 3

Linear Maps

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Section 3AVector Space of Linear Maps[17/0%]

Problem 3A.1

Suppose $b, c \in \mathbb {R}$ . Define $T: \mathbb {R}^{3} \rightarrow \mathbb {R}^{2}$ by

T(x, y, z) = (2 x-4 y+3 z+b, 6 x+c x y z).

Show that $T$ is linear if and only if $b = c = 0$ .

Problem 3A.2

Suppose $b, c \in \mathbb {R}$ . Define $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathbb {R}^{2}$ by

T p = \left(3 p(4)+5 p^{\prime }(6)+b p(1) p(2), \int _{-1}^{2} x^{3} p(x) d x+c \sin p(0)\right).

Show that $T$ is linear if and only if $b = c = 0$ .

Problem 3A.3

Suppose that $T \in \mathcal{L}\left(\mathbf{F}^{n}, \mathbf{F}^{m}\right)$ . Show that there exist scalars $A_{j, k} \in \mathbf{F}$ for $j = 1, \ldots , m$ and $k = 1, \ldots , n$ such that

T\left(x_{1}, \ldots , x_{n}\right) = \left(A_{1,1} x_{1}+\cdots +A_{1, n} x_{n}, \ldots , A_{m, 1} x_{1}+\cdots +A_{m, n} x_{n}\right)

for every $\left(x_{1}, \ldots , x_{n}\right) \in \mathbf{F}^{n}$ . This exercise shows that the linear map $T$ has the form promised in the second to last item of Example 3.3.

Problem 3A.4

Suppose $T \in \mathcal{L}(V, W)$ and $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$ such that $T v_{1}, \ldots , T v_{m}$ is a linearly independent list in $W$ . Prove that $v_{1}, \ldots , v_{m}$ is linearly independent.

Problem 3A.5

Prove that $\mathcal{L}(V, W)$ is a vector space, as was asserted in 3.6.

Problem 3A.6

Prove that multiplication of linear maps has the associative, identity, and distributive properties asserted in 3.8.

Theorem 3.8 (algebraic properties of products of linear maps): associativity: $\left(T_{1} T_{2}\right) T_{3} = T_{1}\left(T_{2} T_{3}\right)$ whenever $T_{1}, T_{2}$ , and $T_{3}$ are linear maps such that the products make sense (meaning $T_{3}$ maps into the domain of $T_{2}$ , and $T_{2}$ maps into the domain of $T_{1}$ ). Identity: $T I = I T = T$ whenever $T \in \mathcal{L}(V, W)$ ; here the first $I$ is the identity operator on $V$ , and the second $I$ is the identity operator on $W$ . Distributive properties: $\left(S_{1}+S_{2}\right) T = S_{1} T+S_{2} T$ and $S\left(T_{1}+T_{2}\right) = S T_{1}+S T_{2}$ whenever $T, T_{1}, T_{2} \in \mathcal{L}(U, V)$ and $S, S_{1}, S_{2} \in \mathcal{L}(V, W)$ .

Problem 3A.7

Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname {dim} V = 1$ and $T \in \mathcal{L}(V)$ , then there exists $\lambda \in \mathbf{F}$ such that $T v = \lambda v$ for all $v \in V$ .

Problem 3A.8

Give an example of a function $\varphi : \mathbb {R}^{2} \rightarrow \mathbb {R}$ such that

\varphi (a v) = a \varphi (v)

for all $a \in \mathbb {R}$ and all $v \in \mathbb {R}^{2}$ but $\varphi$ is not linear. This exercise and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.

Problem 3A.9

Give an example of a function $\varphi : \mathbb {C} \rightarrow \mathbb {C}$ such that

\varphi (w+z) = \varphi (w)+\varphi (z)

for all $w, z \in \mathbb {C}$ but $\varphi$ is not linear. (Here $\mathbb {C}$ is thought of as a complex vector space.)

There also exists a function $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ such that $\varphi$ satisfies the additivity condition above but $\varphi$ is not linear. However, showing the existence of such a function involves considerably more advanced tools.

Problem 3A.10

Prove or give a counterexample: If $q \in \mathcal{P}(\mathbb {R})$ and $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathcal{P}(\mathbb {R})$ is defined by $T p = q \circ p$ , then $T$ is a linear map.

Remark.

The function $T$ defined here differs from the function $T$ defined in the last bullet point of 3.3 by the order of the functions in the compositions. Last bullet point of Example 3.3 (composition): fix a polynomial $q \in \mathcal{P}(\mathbb {R})$ and define a linear map $T \in \mathcal{L}(\mathcal{P}(\mathbb {R}))$ by $(T p)(x) = p(q(x))$ .

Problem 3A.11

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Prove that $T$ is a scalar multiple of the identity if and only if $S T = T S$ for every $S \in \mathcal{L}(V)$ .

Problem 3A.12

Suppose $U$ is a subspace of $V$ with $U \neq V$ . Suppose $S \in \mathcal{L}(U, W)$ and $S \neq 0$ (which means that $S u \neq 0$ for some $u \in U$ ). Define $T: V \rightarrow W$ by

T v = \begin{cases} S v & \text{ if } v \in U, \\ 0 & \text{ if } v \in V \text{ and } v \notin U.\end{cases}

Prove that $T$ is not a linear map on $V$ .

Problem 3A.13

Suppose $V$ is finite-dimensional. Prove that every linear map on a subspace of $V$ can be extended to a linear map on $V$ . In other words, show that if $U$ is a subspace of $V$ and $S \in \mathcal{L}(U, W)$ , then there exists $T \in \mathcal{L}(V, W)$ such that $T u = S u$ for all $u \in U$ .

Remark.

The result in this exercise is used in the proof of 3.125. Theorem 3.125 (dimension of the annihilator): suppose $V$ is finite-dimensional and $U$ is a subspace of $V$ . Then $\operatorname {dim} U^{0} = \operatorname {dim} V-\operatorname {dim} U$ .

Problem 3A.14

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 0$ , and suppose $W$ is infinitedimensional. Prove that $\mathcal{L}(V, W)$ is infinite-dimensional.

Problem 3A.15

Suppose $v_{1}, \ldots , v_{m}$ is a linearly dependent list of vectors in $V$ . Suppose also that $W \neq \left\{ 0\right\}$ . Prove that there exist $w_{1}, \ldots , w_{m} \in W$ such that no $T \in \mathcal{L}(V, W)$ satisfies $T v_{k} = w_{k}$ for each $k = 1, \ldots , m$ .

Problem 3A.16

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 1$ . Prove that there exist $S, T \in \mathcal{L}(V)$ such that $S T \neq T S$ .

Problem 3A.17

Suppose $V$ is finite-dimensional. Show that the only two-sided ideals of $\mathcal{L}(V)$ are $\left\{ 0\right\}$ and $\mathcal{L}(V)$ .

Remark.

A subspace $\mathcal{E}$ of $\mathcal{L}(V)$ is called a two-sided ideal of $\mathcal{L}(V)$ if $T E \in \mathcal{E}$ and $E T \in \mathcal{E}$ for all $E \in \mathcal{E}$ and all $T \in \mathcal{L}(V)$ .

Section 3BNull Spaces and Ranges[33/0%]

Problem 3B.1

Give an example of a linear map $T$ with $\operatorname {dim}\operatorname {null}T = 3$ and $\operatorname {dim}$ range $T = 2$ .

Problem 3B.2

Suppose $S, T \in \mathcal{L}(V)$ are such that range $S \subseteq \operatorname {null}T$ . Prove that $(S T)^{2} = 0$ .

Problem 3B.3

(a)(b)

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$ . Define $T \in \mathcal{L}\left(\mathbf{F}^{m}, V\right)$ by

T\left(z_{1}, \ldots , z_{m}\right) = z_{1} v_{1}+\cdots +z_{m} v_{m}.

(a)

What property of $T$ corresponds to $v_{1}, \ldots , v_{m}$ spanning $V$ ?

(b)

What property of $T$ corresponds to the list $v_{1}, \ldots , v_{m}$ being linearly independent?

Problem 3B.4

Show that $\left\{ T \in \mathcal{L}\left(\mathbb {R}^{5}, \mathbb {R}^{4}\right) \text{ : dim } \operatorname {null} T > 2\right\}$ is not a subspace of $\mathcal{L}\left(\mathbb {R}^{5}, \mathbb {R}^{4}\right)$ .

Problem 3B.5

Give an example of $T \in \mathcal{L}\left(\mathbb {R}^{4}\right)$ such that range $T =\operatorname {null}T$ .

Problem 3B.6

Prove that there does not exist $T \in \mathcal{L}\left(\mathbb {R}^{5}\right)$ such that range $T =\operatorname {null}T$ .

Problem 3B.7

Suppose $V$ and $W$ are finite-dimensional with $2 \leq \operatorname {dim} V \leq \operatorname {dim} W$ . Show that $\left\{ T \in \mathcal{L}(V, W): T\text{ is not injective }\right\}$ is not a subspace of $\mathcal{L}(V, W)$ .

Problem 3B.8

Suppose $V$ and $W$ are finite-dimensional with $\operatorname {dim} V \geq \operatorname {dim} W \geq 2$ . Show that $\left\{ T \in \mathcal{L}(V, W): T\text{ is not surjective }\right\}$ is not a subspace of $\mathcal{L}(V, W)$ .

Problem 3B.9

Suppose $T \in \mathcal{L}(V, W)$ is injective and $v_{1}, \ldots , v_{n}$ is linearly independent in $V$ . Prove that $T v_{1}, \ldots , T v_{n}$ is linearly independent in $W$ .

Problem 3B.10

Suppose $v_{1}, \ldots , v_{n}$ spans $V$ and $T \in \mathcal{L}(V, W)$ . Show that $T v_{1}, \ldots , T v_{n}$ spans range $T$ .

Problem 3B.11

Suppose that $V$ is finite-dimensional and that $T \in \mathcal{L}(V, W)$ . Prove that there exists a subspace $U$ of $V$ such that

U \cap \operatorname {null} T = \left\{ 0\right\} \quad \text{ and } \quad \operatorname {range} T = \left\{ T u: u \in U\right\} .

Problem 3B.12

Suppose $T$ is a linear map from $\mathbf{F}^{4}$ to $\mathbf{F}^{2}$ such that

\operatorname {null} T = \left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbf{F}^{4}: x_{1} = 5 x_{2} \text{ and } x_{3} = 7 x_{4}\right\} .

Prove that $T$ is surjective.

Problem 3B.13

Suppose $U$ is a three-dimensional subspace of $\mathbb {R}^{8}$ and that $T$ is a linear map from $\mathbb {R}^{8}$ to $\mathbb {R}^{5}$ such that $\operatorname {null}T = U$ . Prove that $T$ is surjective.

Problem 3B.14

Prove that there does not exist a linear map from $\mathbf{F}^{5}$ to $\mathbf{F}^{2}$ whose null space equals $\left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \in \mathbf{F}^{5}: x_{1} = 3 x_{2} \text{ and } x_{3} = x_{4} = x_{5}\right\}$ .

Problem 3B.15

Suppose there exists a linear map on $V$ whose null space and range are both finite-dimensional. Prove that $V$ is finite-dimensional.

Problem 3B.16

Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists an injective linear map from $V$ to $W$ if and only if $\operatorname {dim} V \leq \operatorname {dim} W$ .

Problem 3B.17

Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists a surjective linear map from $V$ onto $W$ if and only if $\operatorname {dim} V \geq \operatorname {dim} W$ .

Problem 3B.18

Suppose $V$ and $W$ are finite-dimensional and that $U$ is a subspace of $V$ . Prove that there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname {null}T = U$ if and only if $\operatorname {dim} U \geq \operatorname {dim} V-\operatorname {dim} W$ .

Problem 3B.19

Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$ . Prove that $T$ is injective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $S T$ is the identity operator on $V$ .

Problem 3B.20

Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$ . Prove that $T$ is surjective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $T S$ is the identity operator on $W$ .

Problem 3B.21

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V, W)$ , and $U$ is a subspace of $W$ . Prove that $\left\{ v \in V: T v \in U\right\}$ is a subspace of $V$ and

\operatorname {dim}\left\{ v \in V: T v \in U\right\} = \operatorname {dim}\operatorname {null}T+\operatorname {dim}(U \cap \operatorname {range} T).

Problem 3B.22

Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$ . Prove that

\operatorname {dim}\operatorname {null}S T \leq \operatorname {dim}\operatorname {null}S+\operatorname {dim}\operatorname {null}T

Problem 3B.23

Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$ . Prove that dim range $S T \leq \min \left\{ \operatorname {dim} \operatorname {range} S, \operatorname {dim} \operatorname {range} T\right\}$ .

Problem 3B.24

(a)(b)

(a)

Suppose $\operatorname {dim} V = 5$ and $S, T \in \mathcal{L}(V)$ are such that $S T = 0$ . Prove that dim range $T S \leq 2$ .

(b)

Give an example of $S, T \in \mathcal{L}\left(\mathbf{F}^{5}\right)$ with $S T = 0$ and dim range $T S = 2$ .

Problem 3B.25

Suppose that $W$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$ . Prove that $\operatorname {null}S \subseteq \operatorname {null}T$ if and only if there exists $E \in \mathcal{L}(W)$ such that $T = E S$ .

Problem 3B.26

Suppose that $V$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$ . Prove that range $S \subseteq$ range $T$ if and only if there exists $E \in \mathcal{L}(V)$ such that $S = T E$ .

Problem 3B.27

Suppose $P \in \mathcal{L}(V)$ and $P^{2} = P$ . Prove that $V =\operatorname {null}P \oplus$ range $P$ .

Problem 3B.28

Suppose $D \in \mathcal{L}(\mathcal{P}(\mathbb {R}))$ is such that $\operatorname {deg} D p = (\operatorname {deg} p)-1$ for every nonconstant polynomial $p \in \mathcal{P}(\mathbb {R})$ . Prove that $D$ is surjective.

Remark.

The notation $D$ is used above to remind you of the differentiation map that sends a polynomial p to $p^{\prime }$ .

Problem 3B.29

Suppose $p \in \mathcal{P}(\mathbb {R})$ . Prove that there exists a polynomial $q \in \mathcal{P}(\mathbb {R})$ such that $5 q^{\prime \prime }+3 q^{\prime } = p$ .

Remark.

This exercise can be done without linear algebra, but it's more fun to do it using linear algebra.

Problem 3B.30

Suppose $\varphi \in \mathcal{L}(V, \mathbf{F})$ and $\varphi \neq 0$ . Suppose $u \in V$ is not in $\operatorname {null}\varphi$ . Prove that

V = \operatorname {null} \varphi \oplus \left\{ a u: a \in \mathbf{F}\right\} .

Problem 3B.31

Suppose $V$ is finite-dimensional, $X$ is a subspace of $V$ , and $Y$ is a finitedimensional subspace of $W$ . Prove that there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname {null}T = X$ and range $T = Y$ if and only if $\operatorname {dim} X+\operatorname {dim} Y = \operatorname {dim} V$ .

Problem 3B.32

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 1$ . Show that if $\varphi : \mathcal{L}(V) \rightarrow \mathbf{F}$ is a linear map such that $\varphi (S T) = \varphi (S) \varphi (T)$ for all $S, T \in \mathcal{L}(V)$ , then $\varphi = 0$ .

Problem 3B.33

(a)(b)(c)

Suppose that $V$ and $W$ are real vector spaces and $T \in \mathcal{L}(V, W)$ . Define $T_{\mathrm{C}}: V_{\mathrm{C}} \rightarrow W_{\mathrm{C}}$ by

T_{\mathrm{C}}(u+i v) = T u+i T v

for all $u, v \in V$ . The linear map $T_{\mathrm{C}}$ is called the complexification of the linear map $T$ .

(a)

Show that $T_{\mathrm{C}}$ is a (complex) linear map from $V_{\mathrm{C}}$ to $W_{\mathrm{C}}$ .

(b)

Show that $T_{\mathrm{C}}$ is injective if and only if $T$ is injective.

(c)

Show that range $T_{\mathrm{C}} = W_{\mathrm{C}}$ if and only if range $T = W$ .

Remark.

The complexification $V_{\mathrm{C}}$ was defined in Exercise 8 in Section 1B: suppose $V$ is a real vector space; the complexification of $V$ , denoted by $V_{\mathrm{C}}$ , equals $V \times V$ , with an element $(u,v)$ , $u, v \in V$ , written as $u+iv$ ; addition on $V_{\mathrm{C}}$ is defined by $(u_1+iv_1)+(u_2+iv_2) = (u_1+u_2)+i(v_1+v_2)$ ; complex scalar multiplication on $V_{\mathrm{C}}$ is defined by $(a+bi)(u+iv) = (au-bv)+i(av+bu)$ for all $a, b \in \mathbb {R}$ and all $u, v \in V$ .

Section 3CMatrices[17/0%]

Problem 3C.1

Suppose $T \in \mathcal{L}(V, W)$ . Show that with respect to each choice of bases of $V$ and $W$ , the matrix of $T$ has at least dim range $T$ nonzero entries.

Problem 3C.2

Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$ . Prove that dim range $T = 1$ if and only if there exist a basis of $V$ and a basis of $W$ such that with respect to these bases, all entries of $\mathcal{M}(T)$ equal 1.

Problem 3C.3

(a)(b)

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$ and $w_{1}, \ldots , w_{m}$ is a basis of $W$ .

(a)

Show that if $S, T \in \mathcal{L}(V, W)$ , then $\mathcal{M}(S+T) = \mathcal{M}(S)+\mathcal{M}(T)$ .

(b)

Show that if $\lambda \in \mathbf{F}$ and $T \in \mathcal{L}(V, W)$ , then $\mathcal{M}(\lambda T) = \lambda \mathcal{M}(T)$ .

This exercise asks you to verify 3.35 and 3.38.

Problem 3C.4

Suppose that $D \in \mathcal{L}\left(\mathcal{P}_{3}(\mathbb {R}), \mathcal{P}_{2}(\mathbb {R})\right)$ is the differentiation map defined by $D p = p^{\prime }$ . Find a basis of $\mathcal{P}_{3}(\mathbb {R})$ and a basis of $\mathcal{P}_{2}(\mathbb {R})$ such that the matrix of $D$ with respect to these bases is

\left[\begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]

Compare with Example 3.33. The next exercise generalizes this exercise.

Problem 3C.5

Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$ . Prove that there exist a basis of $V$ and a basis of $W$ such that with respect to these bases, all entries of $\mathcal{M}(T)$ are 0 except that the entries in row $k$ , column $k$ , equal 1 if $1 \leq k \leq \operatorname {dim}$ range $T$ .

Problem 3C.6

Suppose $v_{1}, \ldots , v_{m}$ is a basis of $V$ and $W$ is finite-dimensional. Suppose $T \in \mathcal{L}(V, W)$ . Prove that there exists a basis $w_{1}, \ldots , w_{n}$ of $W$ such that all entries in the first column of $\mathcal{M}(T)$ [with respect to the bases $v_{1}, \ldots , v_{m}$ and $w_{1}, \ldots , w_{n}$ ] are 0 except for possibly a 1 in the first row, first column.

Remark.

In this exercise, unlike Exercise 5, you are given the basis of $V$ instead of being able to choose a basis of $V$ .

Problem 3C.7

Suppose $w_{1}, \ldots , w_{n}$ is a basis of $W$ and $V$ is finite-dimensional. Suppose $T \in \mathcal{L}(V, W)$ . Prove that there exists a basis $v_{1}, \ldots , v_{m}$ of $V$ such that all entries in the first row of $\mathcal{M}(T)$ [with respect to the bases $v_{1}, \ldots , v_{m}$ and $w_{1}, \ldots , w_{n}$ ] are 0 except for possibly a 1 in the first row, first column.

Remark.

In this exercise, unlike Exercise 5, you are given the basis of W instead of being able to choose a basis of W.

Problem 3C.8

Suppose $A$ is an $m$ -by- $n$ matrix and $B$ is an $n$ -by- $p$ matrix. Prove that

(A B)_{j, .} = A_{j, .} B

for each $1 \leq j \leq m$ . In other words, show that row $j$ of $A B$ equals (row $j$ of $A$ ) times $B$ .

Remark.

This exercise gives the row version of 3.48. Theorem 3.48 (column of matrix product equals matrix times column): suppose $A$ is an $m$ -by- $n$ matrix and $B$ is an $n$ -by- $p$ matrix. Then $(A B)_{\cdot , k} = A B_{\cdot , k}$ if $1 \leq k \leq p$ . In other words, column $k$ of $A B$ equals $A$ times column $k$ of $B$ .

Problem 3C.9

Suppose $a = \left(\begin{array}{lll}a_{1} & \cdots & a_{n}\end{array}\right)$ is a 1-by- $n$ matrix and $B$ is an $n$ -by- $p$ matrix. Prove that

a B = a_{1} B_{1, \cdot }+\cdots +a_{n} B_{n, \cdot }

In other words, show that $a B$ is a linear combination of the rows of $B$ , with the scalars that multiply the rows coming from $a$ .

Remark.

This exercise gives the row version of 3.50. Theorem 3.50 (linear combination of columns): suppose $A$ is an $m$ -by- $n$ matrix and $b$ is an $n$ -by- $1$ matrix with entries $b_{1}, \ldots , b_{n}$ . Then $A b = b_{1} A_{\cdot , 1}+\cdots +b_{n} A_{\cdot , n}$ . In other words, $A b$ is a linear combination of the columns of $A$ , with the scalars that multiply the columns coming from $b$ .

Problem 3C.10

Give an example of 2-by-2 matrices $A$ and $B$ such that $A B \neq B A$ .

Problem 3C.11

Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose $A, B, C, D, E$ , and $F$ are matrices whose sizes are such that $A(B+C)$ and $(D+E) F$ make sense. Explain why $A B+A C$ and $D F+E F$ both make sense and prove that

A(B+C) = A B+A C \quad \text{ and } \quad (D+E) F = D F+E F

Problem 3C.12

Prove that matrix multiplication is associative. In other words, suppose $A, B$ , and $C$ are matrices whose sizes are such that $(A B) C$ makes sense. Explain why $A(B C)$ makes sense and prove that

(A B) C = A(B C).

Try to find a clean proof that illustrates the following quote from Emil Artin: "It is my experience that proofs involving matrices can be shortened by 50 if one throws the matrices out."

Problem 3C.13

Suppose $A$ is an $n$ -by- $n$ matrix and $1 \leq j, k \leq n$ . Show that the entry in row $j$ , column $k$ , of $A^{3}$ (which is defined to mean $A A A$ ) is

\sum _{p = 1}^{n} \sum _{r = 1}^{n} A_{j, p} A_{p, r} A_{r, k}

Problem 3C.14

Suppose $m$ and $n$ are positive integers. Prove that the function $A \mapsto A^{\mathrm{t}}$ is a linear map from $\mathbf{F}^{m, n}$ to $\mathbf{F}^{n, m}$ .

Problem 3C.15

Prove that if $A$ is an $m$ -by- $n$ matrix and $C$ is an $n$ -by- $p$ matrix, then

(A C)^{\mathrm{t}} = C^{\mathrm{t}} A^{\mathrm{t}}.

This exercise shows that the transpose of the product of two matrices is the product of the transposes in the opposite order.

Problem 3C.16

Suppose $A$ is an $m$ -by- $n$ matrix with $A \neq 0$ . Prove that the rank of $A$ is 1 if and only if there exist $\left(c_{1}, \ldots , c_{m}\right) \in \mathbf{F}^{m}$ and $\left(d_{1}, \ldots , d_{n}\right) \in \mathbf{F}^{n}$ such that $A_{j, k} = c_{j} d_{k}$ for every $j = 1, \ldots , m$ and every $k = 1, \ldots , n$ .

Problem 3C.17

Suppose $T \in \mathcal{L}(V)$ , and $u_{1}, \ldots , u_{n}$ and $v_{1}, \ldots , v_{n}$ are bases of $V$ . Prove that the following are equivalent.

$T$ is injective.
The columns of $\mathcal{M}(T)$ are linearly independent in $\mathbf{F}^{n, 1}$ .
The columns of $\mathcal{M}(T)$ span $\mathbf{F}^{n, 1}$ .
The rows of $\mathcal{M}(T)$ span $\mathbf{F}^{1, n}$ .
The rows of $\mathcal{M}(T)$ are linearly independent in $\mathbf{F}^{1, n}$ . Here $\mathcal{M}(T)$ means $\mathcal{M}\left(T,\left(u_{1}, \ldots , u_{n}\right),\left(v_{1}, \ldots , v_{n}\right)\right)$ .

Section 3DInvertibility and Isomorphisms[24/0%]

Problem 3D.1

Suppose $T \in \mathcal{L}(V, W)$ is invertible. Show that $T^{-1}$ is invertible and

\left(T^{-1}\right)^{-1} = T.

Problem 3D.2

Suppose $T \in \mathcal{L}(U, V)$ and $S \in \mathcal{L}(V, W)$ are both invertible linear maps. Prove that $S T \in \mathcal{L}(U, W)$ is invertible and that $(S T)^{-1} = T^{-1} S^{-1}$ .

Problem 3D.3

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Prove that the following are equivalent.

$T$ is invertible.
$T v_{1}, \ldots , T v_{n}$ is a basis of $V$ for every basis $v_{1}, \ldots , v_{n}$ of $V$ .
$T v_{1}, \ldots , T v_{n}$ is a basis of $V$ for some basis $v_{1}, \ldots , v_{n}$ of $V$ .

Problem 3D.4

Suppose $V$ is finite-dimensional and $\operatorname {dim} V > 1$ . Prove that the set of noninvertible linear maps from $V$ to itself is not a subspace of $\mathcal{L}(V)$ .

Problem 3D.5

Suppose $V$ is finite-dimensional, $U$ is a subspace of $V$ , and $S \in \mathcal{L}(U, V)$ . Prove that there exists an invertible linear map $T$ from $V$ to itself such that $T u = S u$ for every $u \in U$ if and only if $S$ is injective.

Problem 3D.6

Suppose that $W$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$ . Prove that $\operatorname {null}S =\operatorname {null}T$ if and only if there exists an invertible $E \in \mathcal{L}(W)$ such that $S = E T$ .

Problem 3D.7

Suppose that $V$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$ . Prove that range $S =$ range $T$ if and only if there exists an invertible $E \in \mathcal{L}(V)$ such that $S = T E$ .

Problem 3D.8

Suppose $V$ and $W$ are finite-dimensional and $S, T \in \mathcal{L}(V, W)$ . Prove that there exist invertible $E_{1} \in \mathcal{L}(V)$ and $E_{2} \in \mathcal{L}(W)$ such that $S = E_{2} T E_{1}$ if and only if $\operatorname {dim}\operatorname {null}S = \operatorname {dim}\operatorname {null}T$ .

Problem 3D.9

Suppose $V$ is finite-dimensional and $T: V \rightarrow W$ is a surjective linear map of $V$ onto $W$ . Prove that there is a subspace $U$ of $V$ such that $\left.T\right|_{U}$ is an isomorphism of $U$ onto $W$ .

Remark.

Here $\left.T\right|_{U}$ means the function $T$ restricted to $U$ . Thus $\left.T\right|_{U}$ is the function whose domain is $U$ , with $\left.T\right|_{U}$ defined by $\left.T\right|_{U}(u) = T u$ for every $u \in U$ .

Problem 3D.10

(a)(b)

Suppose $V$ and $W$ are finite-dimensional and $U$ is a subspace of $V$ . Let

\mathcal{E} = \left\{ T \in \mathcal{L}(V, W): U \subseteq \operatorname {null} T\right\}

(a)

Show that $\mathcal{E}$ is a subspace of $\mathcal{L}(V, W)$ .

(b)

Find a formula for $\operatorname {dim} \mathcal{E}$ in terms of $\operatorname {dim} V, \operatorname {dim} W$ , and $\operatorname {dim} U$ .

Problem 3D.11

Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$ . Prove that

S T \text{ is invertible } \Longleftrightarrow S \text{ and } T \text{ are invertible. }

Problem 3D.12

Suppose $V$ is finite-dimensional and $S, T, U \in \mathcal{L}(V)$ and $S T U = I$ . Show that $T$ is invertible and that $T^{-1} = U S$ .

Problem 3D.13

Show that the result in Exercise 12 can fail without the hypothesis that $V$ is finite-dimensional.

Remark.

Exercise 12: suppose $V$ is finite-dimensional and $S, T, U \in \mathcal{L}(V)$ and $STU = I$ . Then $T$ is invertible and $T^{-1} = US$ .

Problem 3D.14

Prove or give a counterexample: If $V$ is a finite-dimensional vector space and $R, S, T \in \mathcal{L}(V)$ are such that $R S T$ is surjective, then $S$ is injective.

Problem 3D.15

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{m}$ is a list in $V$ such that $T v_{1}, \ldots , T v_{m}$ spans $V$ . Prove that $v_{1}, \ldots , v_{m}$ spans $V$ .

Problem 3D.16

Prove that every linear map from $\mathbf{F}^{n, 1}$ to $\mathbf{F}^{m, 1}$ is given by a matrix multiplication. In other words, prove that if $T \in \mathcal{L}\left(\mathbf{F}^{n, 1}, \mathbf{F}^{m, 1}\right)$ , then there exists an $m$ -by- $n$ matrix $A$ such that $T x = A x$ for every $x \in \mathbf{F}^{n, 1}$ .

Problem 3D.17

(a)(b)

Suppose $V$ is finite-dimensional and $S \in \mathcal{L}(V)$ . Define $\mathcal{A} \in \mathcal{L}(\mathcal{L}(V))$ by

\mathcal{A}(T) = S T

for $T \in \mathcal{L}(V)$ .

(a)

Show that $\operatorname {dim}\operatorname {null}\mathcal{A} = (\operatorname {dim} V)(\operatorname {dim}\operatorname {null}S)$ .

(b)

Show that $\operatorname {dim}$ range $\mathcal{A} = (\operatorname {dim} V)(\operatorname {dim}$ range $S)$ .

Problem 3D.18

Show that $V$ and $\mathcal{L}(\mathbf{F}, V)$ are isomorphic vector spaces.

Problem 3D.19

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Prove that $T$ has the same matrix with respect to every basis of $V$ if and only if $T$ is a scalar multiple of the identity operator.

Problem 3D.20

Suppose $q \in \mathcal{P}(\mathbb {R})$ . Prove that there exists a polynomial $p \in \mathcal{P}(\mathbb {R})$ such that

q(x) = \left(x^{2}+x\right) p^{\prime \prime }(x)+2 x p^{\prime }(x)+p(3)

for all $x \in \mathbb {R}$ .

Problem 3D.21

Suppose $n$ is a positive integer and $A_{j, k} \in \mathbf{F}$ for all $j, k = 1, \ldots , n$ . Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables).

The trivial solution $x_{1} = \cdots = x_{n} = 0$ is the only solution to the homogeneous system of equations
$\begin{aligned} & \sum _{k = 1}^{n} A_{1, k} x_{k} = 0 \\ & \vdots \\ & \sum _{k = 1}^{n} A_{n, k} x_{k} = 0 \end{aligned}$
For every $c_{1}, \ldots , c_{n} \in \mathbf{F}$ , there exists a solution to the system of equations
$\begin{gathered} \sum _{k = 1}^{n} A_{1, k} x_{k} = c_{1} \\ \vdots \\ \sum _{k = 1}^{n} A_{n, k} x_{k} = c_{n} \end{gathered}$

Problem 3D.22

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$ . Prove that

\mathcal{M}\left(T,\left(v_{1}, \ldots , v_{n}\right)\right) \text{ is invertible } \Longleftrightarrow T \text{ is invertible. }

Problem 3D.23

Suppose that $u_{1}, \ldots , u_{n}$ and $v_{1}, \ldots , v_{n}$ are bases of $V$ . Let $T \in \mathcal{L}(V)$ be such that $T v_{k} = u_{k}$ for each $k = 1, \ldots , n$ . Prove that

\mathcal{M}\left(T,\left(v_{1}, \ldots , v_{n}\right)\right) = \mathcal{M}\left(I,\left(u_{1}, \ldots , u_{n}\right),\left(v_{1}, \ldots , v_{n}\right)\right).

Problem 3D.24

Suppose $A$ and $B$ are square matrices of the same size and $A B = I$ . Prove that $B A = I$ .

Section 3EProducts and Quotients of Vector Spaces[19/0%]

Problem 3E.1

Suppose $T$ is a function from $V$ to $W$ . The graph of $T$ is the subset of $V \times W$ defined by

\text{ graph of } T = \left\{ (v, T v) \in V \times W: v \in V\right\} .

Prove that $T$ is a linear map if and only if the graph of $T$ is a subspace of $V \times W$ .

Remark.

Formally, a function $T$ from $V$ to $W$ is a subset $T$ of $V \times W$ such that for each $v \in V$ , there exists exactly one element $(v, w) \in T$ . In other words, formally a function is what is called above its graph. We do not usually think of functions in this formal manner. However, if we do become formal, then this exercise could be rephrased as follows: Prove that a function $T$ from $V$ to $W$ is a linear map if and only if $T$ is a subspace of $V \times W$ .

Problem 3E.2

Suppose that $V_{1}, \ldots , V_{m}$ are vector spaces such that $V_{1} \times \cdots \times V_{m}$ is finitedimensional. Prove that $V_{k}$ is finite-dimensional for each $k = 1, \ldots , m$ .

Problem 3E.3

Suppose $V_{1}, \ldots , V_{m}$ are vector spaces. Prove that $\mathcal{L}\left(V_{1} \times \cdots \times V_{m}, W\right)$ and $\mathcal{L}\left(V_{1}, W\right) \times \cdots \times \mathcal{L}\left(V_{m}, W\right)$ are isomorphic vector spaces.

Problem 3E.4

Suppose $W_{1}, \ldots , W_{m}$ are vector spaces. Prove that $\mathcal{L}\left(V, W_{1} \times \cdots \times W_{m}\right)$ and $\mathcal{L}\left(V, W_{1}\right) \times \cdots \times \mathcal{L}\left(V, W_{m}\right)$ are isomorphic vector spaces.

Problem 3E.5

For $m$ a positive integer, define $V^{m}$ by

V^{m} = \underbrace{V \times \cdots \times V}_{m \text{ times }}.

Prove that $V^{m}$ and $\mathcal{L}\left(\mathbf{F}^{m}, V\right)$ are isomorphic vector spaces.

Problem 3E.6

Suppose that $v, x$ are vectors in $V$ and that $U, W$ are subspaces of $V$ such that $v+U = x+W$ . Prove that $U = W$ .

Problem 3E.7

Let $U = \left\{ (x, y, z) \in \mathbb {R}^{3}: 2 x+3 y+5 z = 0\right\}$ . Suppose $A \subseteq \mathbb {R}^{3}$ . Prove that $A$ is a translate of $U$ if and only if there exists $c \in \mathbb {R}$ such that

A = \left\{ (x, y, z) \in \mathbb {R}^{3}: 2 x+3 y+5 z = c\right\} .

Problem 3E.8

(a)(b)

(a)

Suppose $T \in \mathcal{L}(V, W)$ and $c \in W$ . Prove that $\left\{ x \in V: T x = c\right\}$ is either the empty set or is a translate of $\operatorname {null}T$ .

(b)

Explain why the set of solutions to a system of linear equations such as 3.27 is either the empty set or is a translate of some subspace of $\mathbf{F}^{n}$ .

System 3.27: given $A_{j, k} \in \mathbf{F}$ and $c_{1}, \ldots , c_{m} \in \mathbf{F}$ , the system of $m$ linear equations $\sum_{k = 1}^{n} A_{j, k} x_{k} = c_{j}$ , $j = 1, \ldots , m$ , in the $n$ variables $x_{1}, \ldots , x_{n}$ .

Problem 3E.9

Prove that a nonempty subset $A$ of $V$ is a translate of some subspace of $V$ if and only if $\lambda v+(1-\lambda ) w \in A$ for all $v, w \in A$ and all $\lambda \in \mathbf{F}$ .

Problem 3E.10

Suppose $A_{1} = v+U_{1}$ and $A_{2} = w+U_{2}$ for some $v, w \in V$ and some subspaces $U_{1}, U_{2}$ of $V$ . Prove that the intersection $A_{1} \cap A_{2}$ is either a translate of some subspace of $V$ or is the empty set.

Problem 3E.11

(a)(b)

Suppose $U = \left\{ \left(x_{1}, x_{2}, \ldots \right) \in \mathbf{F}^{\infty }: x_{k} \neq 0 \text{ for only finitely many } k\right\}$ .

(a)

Show that $U$ is a subspace of $\mathbf{F}^{\infty }$ .

(b)

Prove that $\mathbf{F}^{\infty } / U$ is infinite-dimensional.

Problem 3E.12

(a)(b)(c)

Suppose $v_{1}, \ldots , v_{m} \in V$ . Let

A = \left\{ \lambda _{1} v_{1}+\cdots +\lambda _{m} v_{m}: \lambda _{1}, \ldots , \lambda _{m} \in \mathbf{F} \text{ and } \lambda _{1}+\cdots +\lambda _{m} = 1\right\} .

(a)

Prove that $A$ is a translate of some subspace of $V$ .

(b)

Prove that if $B$ is a translate of some subspace of $V$ and $\left\{ v_{1}, \ldots , v_{m}\right\} \subseteq B$ , then $A \subseteq B$ .

(c)

Prove that $A$ is a translate of some subspace of $V$ of dimension less than $m$ .

Problem 3E.13

Suppose $U$ is a subspace of $V$ such that $V / U$ is finite-dimensional. Prove that $V$ is isomorphic to $U \times (V / U)$ .

Problem 3E.14

Suppose $U$ and $W$ are subspaces of $V$ and $V = U \oplus W$ . Suppose $w_{1}, \ldots , w_{m}$ is a basis of $W$ . Prove that $w_{1}+U, \ldots , w_{m}+U$ is a basis of $V / U$ .

Problem 3E.15

Suppose $U$ is a subspace of $V$ and $v_{1}+U, \ldots , v_{m}+U$ is a basis of $V / U$ and $u_{1}, \ldots , u_{n}$ is a basis of $U$ . Prove that $v_{1}, \ldots , v_{m}, u_{1}, \ldots , u_{n}$ is a basis of $V$ .

Problem 3E.16

Suppose $\varphi \in \mathcal{L}(V, \mathbf{F})$ and $\varphi \neq 0$ . Prove that $\operatorname {dim} V /(\operatorname {null} \varphi ) = 1$ .

Problem 3E.17

Suppose $U$ is a subspace of $V$ such that $\operatorname {dim} V / U = 1$ . Prove that there exists $\varphi \in \mathcal{L}(V, \mathbf{F})$ such that $\operatorname {null}\varphi = U$ .

Problem 3E.18

(a)(b)

Suppose that $U$ is a subspace of $V$ such that $V / U$ is finite-dimensional.

(a)

Show that if $W$ is a finite-dimensional subspace of $V$ and $V = U+W$ , then $\operatorname {dim} W \geq \operatorname {dim} V / U$ .

(b)

Prove that there exists a finite-dimensional subspace $W$ of $V$ such that $\operatorname {dim} W = \operatorname {dim} V / U$ and $V = U \oplus W$ .

Problem 3E.19

Suppose $T \in \mathcal{L}(V, W)$ and $U$ is a subspace of $V$ . Let $\pi$ denote the quotient map from $V$ onto $V / U$ . Prove that there exists $S \in \mathcal{L}(V / U, W)$ such that $T = S \circ \pi$ if and only if $U \subseteq \operatorname {null}T$ .