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Mathematics LibreTexts

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 \(W_1 = W_2.\)