Confused on notation for matrix

Question

I'm doing linear algebra homework, where's I'm supposed to find the spanning set for the following vector space: $$\{A \in M_2(\mathbb{R}): A\begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} = A\begin{pmatrix} 0 \\ 0 \\ \end{pmatrix}\}$$

What exactly does this notation mean? I'm confused.

Answer 1

If I were to read the notation symbol-by-symbol, I would come up with this:

The set of all $2 \times 2$ matrices $A$ with real entries such that $$ A \pmatrix{1\\2} = A \pmatrix{0\\0} $$

So, for example, $ \left( \begin{smallmatrix}-2&1\\0&0\end{smallmatrix}\right) $ is an element of this set, but $ \left(\begin{smallmatrix} 1&0\\0&1 \end{smallmatrix} \right) $ is not.