Prove that the span of $\{M_1, M_2, M_3\}$ is the set of all symmetric $2\times2$ matrices.

Question

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.

Answer 1

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?

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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?

Answer 2

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 $\{M_1, M_2, M_3\}$

Answer 3

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 $$\begin{bmatrix} a & b \\ b & c\end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} + (c-a)\begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} = aM_1 + (c-a)M_2 + bM_3$$