
# 4.E: Exercises for Chapter 4

#### Calculational Exercises

1. For each of the following sets, either show that the set is a vector space or explain why it is not a vector space.

(a) The set $$\mathbb{R}$$ of real numbers under the usual operations of addition and multiplication.

(b) The set $$\{(x, 0)~ |~ x \in \mathbb{R}\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^2.$$

(c) The set $$\{(x, 1) ~|~ x \in \mathbb{R}\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^2.$$

(d) The set $$\{(x, 0) ~| ~x \in \mathbb{R}, x \geq 0\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^2.$$

(e) The set $$\{(x, 1)~ |~ x \in \mathbb{R}, x \geq 0\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^2.$$

(f) The set $$\left\{ \left[ \begin{array}{cc} a & a+b \\ a+b & a \end{array} \right] ~|~ a, b \in \mathbb{R} \right\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^{2 \times 2}.$$

(g) The set  $$\left\{ \left[ \begin{array}{cc} a & a+b+1 \\ a+b & a \end{array} \right] ~|~ a, b \in \mathbb{R} \right\}$$ under the usual operations of addition and multiplication on $$\mathbb{R}^{2 \times 2}.$$
under the usual operations of addition

2. Show that the space $$V = \{(x_1 , x_2 , x_3 ) \in \mathbb{F}^3 ~|~ x_1 + 2x_2 + 2x_3 = 0\}$$ forms a vector space.

3. For each of the following sets, either show that the set is a subspace of $$\cal{C}(\mathbb{R})$$ or explain why it is not a subspace.

(a) The set $$\{f \in \cal{C}(\mathbb{R}) ~|~ f (x) \leq 0, \forall x \in \mathbb{R}\}.$$

(b) The set $$\{f \in \cal{C}(\mathbb{R}) ~|~ f(0) = 0\}.$$

(c) The set $$\{f \in \cal{C}(\mathbb{R}) ~|~ f (0) = 2\}.$$

(d) The set of all constant functions.

(e) The set $$\{\alpha + \beta sin(x) ~|~ \alpha, \beta \in \mathbb{R}\}.$$

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

5. Let $$\mathbb{F}[z]$$ denote the vector space of all polynomials having coeﬃcient over $$\mathbb{F}$$, and deﬁne $$U$$ to be the subspace of $$\mathbb{F}[z]$$ given by

$U = \{az^2 + bz^5 ~|~ a, b \in \mathbb{F}\}.$

Find a subspace $$W$$ of $$\mathbb{F}[z]$$ such that $$\mathbb{F}[z] = U \oplus W .$$

#### Proof-Writing Exercises

1. Let $$V$$ be a vector space over $$\mathbb{F}$$. Then, given $$a \in \mathbb{F}$$ and $$v \in V$$ such that $$av = 0$$, prove that either $$a = 0$$ or $$v = 0.$$

2. Let $$V$$ be a vector space over $$\mathbb{F},$$ and suppose that $$W_1$$ and $$W_2$$ are subspaces of $$V.$$
Prove that their intersection $$W_1 \cap W_2$$ is also a subspace of $$V.$$

3. Prove or give a counterexample to the following claim:
Claim. Let $$V$$ be a vector space over $$\mathbb{F},$$ and suppose that $$W_1, W_2,$$ and $$W_3$$ are subspaces of $$V$$ such that $$W_1 + W_3 = W_2 + W_3.$$ Then $$W_1 = W_2.$$

4. Prove or give a counterexample to the following claim:
Claim. Let $$V$$ be a vector space over $$\mathbb{F},$$ and suppose that $$W_1 , W_2,$$ and $$W_3$$ are subspaces of $$V$$ such that $$W_1 \oplus W_3 = W_2 \oplus W_3.$$ Then $$W_1 = W_2.$$

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