5.5. Vector spaces exercises

Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.

Exercise 5.1

Prove that the axioms of the vector space \(\mathbb{R}^3\) hold.

Exercise 5.2

Using the subspace condition, determine whether the following subsets of \(\mathbb{R}^3\) are subspaces:

(a)   \(U = \{ (x, y, 0) : x, y \in \mathbb{R} \}\)

(b)   \(V = \{ (1, 2, 0) \}\)

(c)   \(W = \{ (0, y, 0) : y \in \mathbb{R} \}\)

(d)   \(X = \{ (x, y, z) : y = |x|, x, y, z \in \mathbb{R}\}\)

Exercise 5.3

Which of the following sets are subspaces of \(M_2(\mathbb{R})\)?

(a)   \(A = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \mathbb{R}, a + b = 1, \right\}\)

(b)   \(B = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}: a, b, c, d \in \mathbb{R}, a = c = d \right\}\)

 

(c)   \(C = \{ A \in M_{2\times 2} : A^2 = A \}\)

Exercise 5.4

Prove that \(\left\{ (1, 2, 0), (0, 5, 7), (-1, 1, 3) \right\}\) is a basis for \(\mathbb{R}^3\) and represent the vectors \((0, 13, 17)\) and \((2, 3, 1)\) with respect to this basis.

Exercise 5.5

Extend \(\{ (1, 1, 2, 4), (2, -1, -5, 2)\}\) to a basis of \(\mathbb{R}^4\).

Exercise 5.6

Suppose that \(W = \{\vec{u}, \vec{v}, \vec{w}, \vec{x}, \vec{y}\}\) is a subspace of \(\mathbb{R}^n\) where

\[\begin{split} \begin{align*} \vec{u} &= \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}, & \vec{v} &= \begin{pmatrix} 1 \\ -1 \\ 2 \\ 0 \end{pmatrix}, & \vec{w} &= \begin{pmatrix} 2 \\ 1 \\ 5 \\ 3 \end{pmatrix}, & \vec{x} &= \begin{pmatrix} 1 \\ -1 \\ 0 \\ 3 \end{pmatrix}, & \vec{y} &= \begin{pmatrix} 1 \\ -1 \\ 0 \\ 4 \end{pmatrix}. \end{align*} \end{split}\]

Find a basis for \(W\) and determine \(\dim(W)\).