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Any matrix can be expressed as a linear combination of basis matrices, which play the role of the [[basis vector]]s for the matrix space. For example, for the set of 2×2 matrices over the field of real numbers, ''M''<sub>22</sub>(ℝ), one legitimate basis set of matrices is:
:<math>\begin{pmatrix}1&0\\0&0\end{pmatrix}\,,\quad
\begin{pmatrix}0&1\\0&0\end{pmatrix}\,,\quad
\begin{pmatrix}0&0\\1&0\end{pmatrix}\,,\quad
\begin{pmatrix}0&0\\0&1\end{pmatrix}\,,</math>
because any 2×2 matrix can be expressed as:
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