Find a basis for the following subspace of $M_{2\times 2}(\mathbb R) $

Question

$$W=\left\{\begin{bmatrix} a & b \\ c & d \\\end{bmatrix} \in M_{2\times 2}(\mathbb R) \mid a + 2b = c - 3d = 0\right\}$$

This is supposed to be a not so difficult question but it's getting difficult for me to wrap my head around it.

Answer 1

You can write an arbitrary element of the subspace $W$ as

$$\begin{bmatrix} - 2b & b \\ 3d & d \end{bmatrix} = b \begin{bmatrix} -2 &1 \\ 0 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 \\ 3 & 1 \end{bmatrix} $$

by noting $c=3d, a = -2b$.

Note that these two matrices are linearly independent, so $\left\{\begin{bmatrix} -2 &1 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 0 & 0 \\ 3 & 1 \end{bmatrix} \right\} $ form a basis.