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!
To see that $\displaystyle U$ is a subspace of $\displaystyle M_n(\mathbb{R})$ (or $\displaystyle M_n(\mathbb{C})$), 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.