1. ## Subspace proof

Let U be the set of symmetric 2 x 2 matrices. Prove that U is a subspace of the vector space of all 2 x 2 matrices; Find a "standard basis" for U and the dimension of U.

Any help with this would be much appreciated!

2. ## Re: Subspace proof

To see that $U$ is a subspace of $M_n(\mathbb{R})$ (or $M_n(\mathbb{C})$), check that: the zero matrix is symmetric, the sum of two symmetric matrices are symmetric, and that the multiplication of a symmetric by a scalar is still symmetric.

I'm afraid I can't give a hint for a basis without giving away the answer. You have to think about this one yourself, but it is not hard to see what is has to be.