From Linear Algebra by Friedberg, Insel, and Spence:
Given $M_1=\begin{pmatrix} 1&0\\0 &1\end{pmatrix}$, $M_2=\begin{pmatrix} 0&0\\0 &1\end{pmatrix}$ and $M_3=\begin{pmatrix}0&1\\1 &0\end{pmatrix}$,
prove that $\text{span}\{M_1, M_2, M_3\}$ is the set of all symmetric $2 \times2$ matrices.
For reference, we just learned about linear combinations/span, but only in terms of vectors, nothing really with matrices.
Best Answer
Since $M_1, M_2$, and $M_3$ are all symmetric, every matrix in their span will be symmetric, so we need to show that every symmetric matrix is in their span. Every $2 × 2$ symmetric matrix has the form $$M=\begin{pmatrix} a & b \\ b & c \end{pmatrix}$$and since we can write $M = aM_1 + cM_2 + bM_3$, any such $M$ is in the span of $\{M_1, M_2, M_3\}$