1   For each of the following subsets of $\mathbf{F}^3$, determine whether it is a subspace of $\mathbf{F}^3$.
(a)   $\{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 + 2x_2 + 3x_3 = 0 \}$
(b)   $\{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 + 2x_2 + 3x_3 = 4 \}$
(c)   $\{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1x_2x_3 = 0 \}$
(d)   $\{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 = 5x_3 \}$

Solution

Recall that three conditions are necessary and sufficient to show that a subset $U$ of a vector space $V$ over a field $\mathbf{F}$ is a subspace:

1. additive identity

$$0 \in U$$

2. closed under addition

$$u, w \in U \implies u + w \in U$$

3. closed under scalar multiplication

$$a \in \mathbf{F} \land a \in U \implies au \in U$$

(a)   $U = \{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 + 2x_2 + 3x_3 = 0 \}$

Clearly $0 = (0,0,0) \in U$ as $0 + 2 \times 0 + 3 \times 0 = 0$ so 1. additive identity holds.

Let $u = (u_1, u_2, u_3), w = (w_1, w_2, w_3) \in U$. Then

$$(u_1 + w_1) + 2(u_2 + w_2) + 3(u_3 + w_3) \\ = (u_1 + 2u_2 + 3u_3) + (w_1 + 2w_2 + 3w_3) = 0 + 0 = 0,$$

so 2. closed under addition holds.

Let $a \in \mathbf{F}$. Then

$$(au_1) + 2(au_2) + 3(au_3) = a (u_1 + 2u_2 + 3u_3) = a \cdot 0 = 0,$$

so $au = (au_1, au_2, au_3) \in U$, so 3. closed under scalar multiplication holds.

We conclude that $U$ is a subspace. $\quad \square$

(b)   $U = \{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 + 2x_2 + 3x_3 = 4 \}$

$U$ is not a subspace because it doesn't contain $0$. $\quad \square$

(c)   $U = \{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1x_2x_3 = 0 \}$

$U$ is not a subspace because it isn't closed under addition. $(1, 0, 1) \in U$ and $(0,1,0) \in U$ but $(1,0,1) + (0,1,0) = (1,1,1) \notin U$. $\quad \square$

(d)   $U = \{ (x_1,x_2,x_3) \in \mathbf{F}^3: x_1 = 5x_3 \}$

Clearly $0 \in U$ so 1. additive identity holds.

If $u = (u_1, u_2, u_3), w = (w_1, w_2, w_3) \in U$. Then

$$u_1 + w_1 = 5u_3 + 5w_3 = 5(u_3 + w_3) = 5 \cdot 0 = 0,$$

so 2. closed under addition holds.

If $a \in \mathbf{F}$ then

$$au_1 = a(5u_3) = 5(au_3),$$

so $au \in \mathbf{F}$, and therefore 3. closed under scalar multiplication holds.

We conclude that $U$ is a subspace. $\quad \square$

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2   Verify the following assertions about subspaces:

$\text{(a)} \>$ If $b \in \mathbf{F}$, then

$$\{(x_1,x_2,x_3,x_4) \in \mathbf{F}^4: x_3 = 5x_4 + b\}$$

is a subspace of $\mathbf{F}^4$ if and only if $b=0$.

$\text{(b)} \>$ The set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbf{R}^{[0,1]}$.

$\text{(c)} \>$ The set of differentiable real-valued functions on $\mathbf{R}$ is a subspace of $\mathbf{R}^\mathbf{R}$.

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

$\text{(e)} \>$ The set of all sequences of complex numbers with limit $0$ is a subspace of $\mathbf{C}^\infty$.

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

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4   Suppose $b \in \mathbf{R}$. 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 $\mathbf{R}^{[0,1]}$ if and only if $b=0$.

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

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6   (a) Is $\{(a,b,c) \in \mathbf{R}^3: a^3 = b^3\}$ a subspace of $\mathbf{R}^3$?
(b) Is $\{(a,b,c) \in \mathbf{C}^3: a^3=b^3\}$ a subspace of $\mathbf{C}^3$?

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7   Prove or give a counterexample: If $U$ is a nonempty subset of $\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), then $U$ is not a subspace of $\mathbf{R}^2$.

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

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

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

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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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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.

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.

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14   Suppose

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

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

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

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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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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

$$(V_1 + V_2) + V_3 = V_1 + (V_2 + V_3)?$$

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

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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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20   Suppose

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

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

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21   Suppose

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

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

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22   Suppose

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

Find three subspaces $W_1$, $W_2$, $W_3$ of $\mathbf{F}^5$, none of which equals $\{0\}$, such that $\mathbf{F}^5 = U \oplus W_1 \oplus W_2 \oplus W_3$.

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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 $V_1 = V_2$.

Hint: When trying to discover whether a conjecture in linear algebra is true or false, it is often useful to start by experimenting in $\mathbf{F}^2$.

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24   A function $f: \mathbf{R} \to \mathbf{R}$ is called even if

$$f(-x) = f(x)$$

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

$$f(-x) = -f(x)$$

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

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