Vector spaces & bases

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   The vectors $b_1 = \begin{pmatrix} -1 \\ -2 \end{pmatrix}$ and $b_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ form a basis of $\mathbb R^2$. The vector $w = \begin{pmatrix} -6 \\ -9 \end{pmatrix}$ can be written uniquely as $w = k_1 b_1 + k_2 b_2$. Find $k_1$ and $k_2$, the coordinates of $w$ relative to this basis.

2. Warm-up

   The vectors $b_1 = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$ and $b_2 = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$ form a basis of $\mathbb R^2$. The vector $w = \begin{pmatrix} 7 \\ -1 \end{pmatrix}$ can be written uniquely as $w = k_1 b_1 + k_2 b_2$. Find $k_1$ and $k_2$, the coordinates of $w$ relative to this basis.

3. Warm-up

   The vectors $b_1 = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and $b_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ form a basis of $\mathbb R^2$. The vector $w = \begin{pmatrix} -1 \\ 0 \end{pmatrix}$ can be written uniquely as $w = k_1 b_1 + k_2 b_2$. Find $k_1$ and $k_2$, the coordinates of $w$ relative to this basis.

Problems shown under their original terms: Generated (template, solver-computed).