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!

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- Apr 11th 2013, 06:06 PMwidenerl194Subspace 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!

- Apr 11th 2013, 08:13 PMGusbobRe: Subspace proof
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.