Unit 4 test: Vector Spaces

Which set is a subspace of $\mathbb{R}^3$?

Which set is a subspace of $\mathbb{P}_2$, the polynomials of degree at most $2$?

Let $A = \begin{bmatrix} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & 0 \end{bmatrix}$. Find the vector $\mathbf{x} = (x_1, x_2, x_3, x_4)$ in $\operatorname{Nul} A$ with $x_2 = 0$ and $x_4 = 1$.

Let $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \\ 2 & 3 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 3 \\ 1 \\ h \end{bmatrix}$. For what value of $h$ is $\mathbf{b}$ in $\operatorname{Col} A$?

Let $A = \begin{bmatrix} 1 & 3 & 1 \\ 2 & 6 & 0 \\ 1 & 3 & 2 \end{bmatrix}$. Which set is a basis for $\operatorname{Col} A$?

Which set is a basis for $\mathbb{R}^3$?

Let $\mathcal{B} = \lbrace 1,\; 1 + t,\; 1 + t + t^2 \rbrace$, a basis for $\mathbb{P}_2$. Find the coordinate vector $[p]_{\mathcal{B}}$ of $p(t) = 4 + 3t + 2t^2$.

Let $H$ be the set of vectors $(x_1, x_2, x_3, x_4)$ in $\mathbb{R}^4$ with $x_1 - x_2 + 2x_3 = 0$ and $x_2 + x_4 = 0$. Find $\dim H$.

Find the dimension of the vector space of all symmetric $3 \times 3$ matrices (matrices with $A^T = A$).

Find the rank of $\begin{bmatrix} 1 & 2 & 0 & 1 \\ 2 & 4 & 1 & 3 \\ 3 & 6 & 1 & 4 \end{bmatrix}$.

A $6 \times 9$ matrix $A$ has $\dim \operatorname{Nul} A = 4$. Find $\dim \operatorname{Nul} A^T$.

Let $A$ be a $4 \times 6$ matrix. Which statement must be true?

Let $\mathcal{B} = \lbrace (1, 1),\; (1, -1) \rbrace$. Find $[\mathbf{x}]_{\mathcal{B}}$ for $\mathbf{x} = (5, 1)$.

Let $\mathcal{B} = \lbrace \mathbf{b}_1, \mathbf{b}_2 \rbrace$ with $\mathbf{b}_1 = (2, 1)$, $\mathbf{b}_2 = (1, 1)$, and $\mathcal{C} = \lbrace \mathbf{c}_1, \mathbf{c}_2 \rbrace$ with $\mathbf{c}_1 = (1, 0)$, $\mathbf{c}_2 = (1, 1)$. Find the change-of-basis matrix $\underset{\mathcal{C} \leftarrow \mathcal{B}}{P}$, and use it to find $[\mathbf{x}]_{\mathcal{C}}$ when $[\mathbf{x}]_{\mathcal{B}} = (3, -2)$.