Problem Set 6: Vector Spaces and Bases#
Problem 1: Vector Space Verification#
Determine whether each of the following sets forms a vector space under the standard operations of addition and scalar multiplication. If not, identify which axiom(s) fail.
(a) \(V = \{[x\hspace{1em}y]^T \in \mathbb{R}^2 : x \geq 0\}\)
(b) \(V = \{[x\hspace{1em}y\hspace{1em}z]^T \in \mathbb{R}^3 : 2x - y + 3z = 0\}\)
(c) \(V = \{f : \mathbb{R} \to \mathbb{R} : f(0) = 1\}\) with pointwise addition and scalar multiplication
(d) \(V = \{A \in M_{2 \times 2}(\mathbb{R}) : A^T = A\}\) (the set of \(2 \times 2\) symmetric matrices)
Problem 2: Vector Membership#
For each vector space \(V\) and vector \(\mathbf{v}\), determine whether \(\mathbf{v} \in V\).
(a) \(V = \{(x, y, z) \in \mathbb{R}^3 : x + 2y - z = 0\}\), \(\mathbf{v} = (1, -1, -1)\)
(b) \(V = \text{span}\left\{\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ -1 \end{bmatrix}\right\}\), \(\mathbf{v} = \begin{bmatrix} 3 \\ 7 \\ 2 \end{bmatrix}\)
(c) \(V = \{p(x) \in P_3 : p(1) = 0\}\) (polynomials of degree at most 3), \(\mathbf{v} = 2x^3 - 3x^2 + x\)
Problem 3: Subspace Verification#
Determine whether each of the following subsets is a subspace of the given vector space.
(a) \(W = \left\{\begin{bmatrix} a \\ b \\ c \end{bmatrix} : a + b + c = 0\right\}\) in \(\mathbb{R}^3\)
(b) \(W = \left\{\begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} : a = 2b \text{ and } c = d^2\right\}\) in \(\mathbb{R}^4\)
(c) \(W = \{p(x) \in P_2 : p'(0) = 0\}\) in \(P_2\) (polynomials of degree at most 2)
(d) \(W = \{A \in M_{3 \times 3}(\mathbb{R}) : \text{det}(A) = 0\}\) in \(M_{3 \times 3}(\mathbb{R})\)
Problem 4: Orthogonal Subspaces#
(a) Determine whether the following subspaces of \(\mathbb{R}^4\) are orthogonal:
(b) Let \(W_1 = \left\{\begin{bmatrix} x \\ y \\ z \end{bmatrix} : x + y + z = 0\right\}\) and \(W_2 = \text{span}\left\{\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\right\}\) in \(\mathbb{R}^3\). Are \(W_1\) and \(W_2\) orthogonal subspaces?
Problem 5: Basis and Spanning Sets#
(a) Determine whether \(S = \left\{\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\right\}\) is a basis for \(\mathbb{R}^3\).
(b) Does \(S = \{1 + x, 1 - x, x + x^2, 2 + x^2\}\) span \(P_2\)? Is it a basis for \(P_2\)?
(c) Determine whether the set \(S = \left\{\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}\right\}\) spans \(\mathbb{R}^4\). If not, describe the subspace it spans.
(d) Find a basis for the subspace \(W = \left\{\begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix} : a - 2b + c = 0 \text{ and } b + d = 0\right\}\) of \(\mathbb{R}^4\).
Problem 6: Column Space and Nullspace#
For each matrix below, find:
(i) A basis for the column space \(\text{Col}(A)\)
(ii) A basis for the nullspace \(\text{Null}(A)\)
(iii) The dimensions of both spaces
(a) \(A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 1 & 3 & 5 & 7 \end{bmatrix}\)
(b) \(A = \begin{bmatrix} 1 & 0 & 2 & -1 \\ 2 & 1 & 3 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & -1 & 1 & -2 \end{bmatrix}\)
(c) \(A = \begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 1 & 3 & 1 & 3 \\ 2 & 5 & 1 & 4 \end{bmatrix}\)
Problem 7: Challenge Problem#
Let \(V = \{(x, y, z) \in \mathbb{R}^3 : x + y + 2z = 0\}\) and let \(W = \text{span}\left\{\begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ -1 \end{bmatrix}\right\}\).
(a) Show that \(W \subseteq V\).
(b) Is \(W = V\)? Justify your answer.
(c) Find a basis for \(W^\perp\) (the orthogonal complement of \(W\) in \(\mathbb{R}^3\)).