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) \(U = \{ (x, y, 0)^\mathsf{T} : x, y \in \mathbb{R} \}\)
(b) \(V = \{ (1, 2, 0)^\mathsf{T} \}\)
(c) \(W = \{ (0, y, 0)^\mathsf{T} : y \in \mathbb{R} \}\)
(d) \(X = \{ (x, y, z)^\mathsf{T} : y = |x|, x, y, z \in \mathbb{R}\}\)
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 \(\left\{ \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 5 \\ 7 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix} \right\}\) is a basis for \(\mathbb{R}^3\) and represent the vectors \(\begin{pmatrix} 0 \\ 13 \\ 17 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}\) with respect to this basis.
Exercise 5.5
Extend \(\left\{ \begin{pmatrix} 1 \\ 1 \\ 2 \\ 4 \end{pmatrix}, \begin{pmatrix} 2 \\ -1 \\ -5 \\ 2 \end{pmatrix} \right\}\) to a basis of \(\mathbb{R}^4\).
Exercise 5.6
Suppose that \(W\) is a subspace of \(\mathbb{R}^n\) generated by the vectors
Find a basis for \(W\) and determine \(\dim(W)\).