# Describe the smallest subspace of $M_{2\times 2}$ that contains matrices…

Describe the smallest subspace of $M_{2\times 2}$ that contains matrices

$$\begin{bmatrix}2&1\\0&0\end{bmatrix},\begin{bmatrix}1&0\\0&2\end{bmatrix},\begin{bmatrix}0&-1\\0&0\end{bmatrix}\;.$$

Find the dimension of this subspace.

It sounds like I would find the basis. I know how to do this with vectors, but how do I do this with a set of matrices?

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$M_{2\times 2}$ is isomorphic to $\Bbb R^4$ by the correspondence $$\begin{bmatrix}a&b\\c&d\end{bmatrix}\leftrightarrow\begin{bmatrix}a\\b\\c\\d\end{bmatrix}\;;$$ treat the matrices as $4$-vectors, just displayed in a different way.
@LearningPython: I’ve not done the row reduction to check, but that could well be right; certainly there’s nothing obviously wrong with it, and it reduces further to $\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}$, which I know is right. What it shows is that the original three matrices are linearly independent, so what do you conclude? – Brian M. Scott Nov 19 '12 at 8:07