Prove that the span of $\{M_1, M_2, M_3\}$ is the set of all symmetric $2\times2$ matrices. 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.
 A: 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 $\{M_1, M_2, M_3\}$
A: HINT
Span is the set of all linear combinations. Note that
$$
aM_1 + bM_2 + cM_3 =
\begin{pmatrix}
a & c \\
c & a+b
\end{pmatrix}
$$
Can you finish finding the span?

HINT 2
The way you phrased it, the span does not include all symmetric matrices, but the span is that set. Is $M_2$ symmetric?
A: Clearly $M_1, M_2$ and $M_3$ are symmetric.
Conversely, for an arbitrary symmetric matrix $\begin{bmatrix} a & b \\ b & c\end{bmatrix}$ we have
$$\begin{bmatrix} a & b \\ b & c\end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} + (c-a)\begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} = aM_1 + (c-a)M_2 + bM_3$$
