Math Problems Vault

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 $V \times W$ .

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Remark.

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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 $k = 1, \ldots , m$ .

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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.

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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.

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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.

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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 $U = W$ .

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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\} .

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Problem 3E.8

(a)(b)

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(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 $\lambda \in \mathbf{F}$ .

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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.

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Problem 3E.11

(a)(b)

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

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(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\} .

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(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 $U \times (V / U)$ .

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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 $V / U$ .

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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 $V$ .

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Problem 3E.16

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

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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 $\operatorname {null}\varphi = U$ .

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Problem 3E.18

(a)(b)

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

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(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 $U \subseteq \operatorname {null}T$ .

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