Thread: Subspace of 3x3 matrices containing symmetric and lower-triangular matices

What is the smallest subspace of 3 by 3 matrices that contains all symmetric matrices and all lower matrices? What is the largest subspace that is contained in both of those subspaces?

I'm really not sure how to even approach this problem.

The trivial subspace, which consists of a 3x3 null-matrix is the smallest subspace of the vector space of all symmetric and lower 3x3 matrices, since it contains only one element, the 3x3 null-matrix, which satisfies both of the required conditions.