Math Problems Vault

Problem 1C.1

(a)(b)(c)(d)

For each of the following subsets of $\mathbf{F}^{3}$ , determine whether it is a subspace of $\mathbf{F}^{3}$ .

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(a)

$\left\{ \left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}+2 x_{2}+3 x_{3} = 0\right\}$

(b)

$\left\{ \left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}+2 x_{2}+3 x_{3} = 4\right\}$

(c)

$\left\{ \left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1} x_{2} x_{3} = 0\right\}$

(d)

$\left\{ \left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1} = 5 x_{3}\right\}$

Problem 1C.2

Verify all assertions about subspaces in Example 1.35.

Example 1.35:

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(a)

If $b \in \mathbf{F}$ , then $\left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbf{F}^{4}: x_{3} = 5 x_{4}+b\right\}$ is a subspace of $\mathbf{F}^{4}$ if and only if $b = 0$ .

(b)

The set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbb {R}^{[0,1]}$ .

(c)

The set of differentiable real-valued functions on $\mathbb {R}$ is a subspace of $\mathbb {R}^{\mathbb {R}}$ .

(d)

The set of differentiable real-valued functions $f$ on the interval $(0,3)$ such that $f^{\prime }(2) = b$ is a subspace of $\mathbb {R}^{(0,3)}$ if and only if $b = 0$ .

(e)

The set of all sequences of complex numbers with limit 0 is a subspace of $\mathbb {C}^{\infty }$ .

Problem 1C.3

Show that the set of differentiable real-valued functions $f$ on the interval $(-4,4)$ such that $f^{\prime }(-1) = 3 f(2)$ is a subspace of $\mathbb {R}^{(-4,4)}$ .

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Problem 1C.4

Suppose $b \in \mathbb {R}$ . Show that the set of continuous real-valued functions $f$ on the interval $[0,1]$ such that $\int_{0}^{1} f = b$ is a subspace of $\mathbb {R}^{[0,1]}$ if and only if $b = 0$ .

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Problem 1C.5

Is $\mathbb {R}^{2}$ a subspace of the complex vector space $\mathbb {C}^{2}$ ?

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Problem 1C.6

(a)(b)

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(a)

Is $\left\{ (a, b, c) \in \mathbb {R}^{3}: a^{3} = b^{3}\right\}$ a subspace of $\mathbb {R}^{3}$ ?

(b)

Is $\left\{ (a, b, c) \in \mathbb {C}^{3}: a^{3} = b^{3}\right\}$ a subspace of $\mathbb {C}^{3}$ ?

Problem 1C.7

Prove or give a counterexample: If $U$ is a nonempty subset of $\mathbb {R}^{2}$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$ ), then $U$ is a subspace of $\mathbb {R}^{2}$ .

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Problem 1C.8

Give an example of a nonempty subset $U$ of $\mathbb {R}^{2}$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbb {R}^{2}$ .

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Problem 1C.9

A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x+p)$ for all $x \in \mathbb {R}$ . Is the set of periodic functions from $\mathbb {R}$ to $\mathbb {R}$ a subspace of $\mathbb {R}^{\mathbb {R}}$ ? Explain.

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Problem 1C.10

Suppose $V_{1}$ and $V_{2}$ are subspaces of $V$ . Prove that the intersection $V_{1} \cap V_{2}$ is a subspace of $V$ .

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Problem 1C.11

Prove that the intersection of every collection of subspaces of $V$ is a subspace of $V$ .

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Problem 1C.12

Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.

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Problem 1C.13

Prove that the union of three subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces contains the other two.

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Remark.

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This exercise is surprisingly harder than Exercise 12, possibly because this exercise is not true if we replace $\mathbf{F}$ with a field containing only two elements.

Problem 1C.14

Suppose

U = \left\{ (x,-x, 2 x) \in \mathbf{F}^{3}: x \in \mathbf{F}\right\} \quad \text{ and } \quad W = \left\{ (x, x, 2 x) \in \mathbf{F}^{3}: x \in \mathbf{F}\right\} .

Describe $U+W$ using symbols, and also give a description of $U+W$ that uses no symbols.

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Problem 1C.15

Suppose $U$ is a subspace of $V$ . What is $U+U$ ?

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Problem 1C.16

Is the operation of addition on the subspaces of $V$ commutative? In other words, if $U$ and $W$ are subspaces of $V$ , is $U+W = W+U$ ?

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Problem 1C.17

Is the operation of addition on the subspaces of $V$ associative? In other words, if $V_{1}, V_{2}, V_{3}$ are subspaces of $V$ , is

\left(V_{1}+V_{2}\right)+V_{3} = V_{1}+\left(V_{2}+V_{3}\right)?

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Problem 1C.18

Does the operation of addition on the subspaces of $V$ have an additive identity? Which subspaces have additive inverses?

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Problem 1C.19

Prove or give a counterexample: If $V_{1}, V_{2}, U$ are subspaces of $V$ such that

V_{1}+U = V_{2}+U,

then $V_{1} = V_{2}$ .

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Problem 1C.20

Suppose

U = \left\{ (x, x, y, y) \in \mathbf{F}^{4}: x, y \in \mathbf{F}\right\} .

Find a subspace $W$ of $\mathbf{F}^{4}$ such that $\mathbf{F}^{4} = U \oplus W$ .

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Problem 1C.21

Suppose

U = \left\{ (x, y, x+y, x-y, 2 x) \in \mathbf{F}^{5}: x, y \in \mathbf{F}\right\} .

Find a subspace $W$ of $\mathbf{F}^{5}$ such that $\mathbf{F}^{5} = U \oplus W$ .

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Problem 1C.22

Suppose

U = \left\{ (x, y, x+y, x-y, 2 x) \in \mathbf{F}^{5}: x, y \in \mathbf{F}\right\} .

Find three subspaces $W_{1}, W_{2}, W_{3}$ of $\mathbf{F}^{5}$ , none of which equals $\left\{ 0\right\}$ , such that $\mathbf{F}^{5} = U \oplus W_{1} \oplus W_{2} \oplus W_{3}$ .

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Problem 1C.23

Prove or give a counterexample: If $V_{1}, V_{2}, U$ are subspaces of $V$ such that

V = V_{1} \oplus U \quad \text{ and } \quad V = V_{2} \oplus U,

then $V_{1} = V_{2}$ .

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Problem 1C.24

A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called even if

f(-x) = f(x)

for all $x \in \mathbb {R}$ . A function $f: \mathbb {R} \rightarrow \mathbb {R}$ is called odd if

f(-x) = -f(x)

for all $x \in \mathbb {R}$ . Let $V_{\mathrm{e}}$ denote the set of real-valued even functions on $\mathbb {R}$ and let $V_{\mathrm{o}}$ denote the set of real-valued odd functions on $\mathbb {R}$ . Show that $\mathbb {R}^{\mathbb {R}} = V_{\mathrm{e}} \oplus V_{\mathrm{o}}$ .

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