I'm not really sure how to show this. It's the part about symmetric matrices that throws me off.Show that the set spans the subspace of symmetric matrices in .

What I know:

A symmetric matrix has the property that . To show that the set spans, I could create a matrix and show that if there are leading ones in each row, without a pivot in the augmented column, the vectors will span the subspace:

Therefore, looking at the RREF of this matrix, I'll end up with a pivot in the augmented column, which doesn't lead to the right conclusion.

As I said before, the condition is throwing me off, since showing something like the fact that two vectors in span the subspace of , where S is the set of all vectors whose first and last components are zero, is relatively straightforward.