Symmetric $3\times 3$ Matrices - Base matrix

Let $V$ be the set of all symmetric $3 \times 3$ matrices. (Recall that $V$ is a subspace of $M_{3\times 3}.$) Find a basis of $V$ , and show that it is a basis.

I'm not sure what the question means by "basis of $V$." Any help would be appreciated.

Solution 1:

Note that a $3\times 3$ symmetric matrix can be written as: $$ \begin{bmatrix} a&d&e\\ d&b&f\\ e&f&c \end{bmatrix} $$ That can be docomposed as a sum:

$$ a\begin{bmatrix} 1&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}+ b\begin{bmatrix} 0&0&0\\ 0&1&0\\ 0&0&0 \end{bmatrix}+ c\begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix}+ d\begin{bmatrix} 0&1&0\\ 1&0&0\\ 0&0&0 \end{bmatrix}+ e\begin{bmatrix} 0&0&1\\ 0&0&0\\ 1&0&0 \end{bmatrix}+ f\begin{bmatrix} 0&0&0\\ 0&0&1\\ 0&1&0 \end{bmatrix} $$

It is not difficult to show that the six matrices in this decomposition are linearly independent, so these are a basis for the subspace of symmetric matrices.