Question

Find a basis for the space V of real nxn symmetric matrices. Prove your claim. Find a basis for the complementary space, that is, the vector space W such that every nxn real matrix is a sum of a vector vEV and wEW, and that V intersection W is zero vector

Answer 1

"Find a basis for the space V of real nxn symmetric matrices. Prove your claim."

Can you describe a basis for ALL nxn matrices? If you can, and you can justify why it should give you an idea of how the basis for symmetric matrices should look like.

Answer 2

what about the second part of the question

Answer 3

Once you have a basis for symmetric matrices and the whole nxn case you can figure out the dimension of the complementary space.

Let's say you know that the symmetric case has k, the complementary l and the whole space nxn dimension. There's a theorem that says k+l = nxn. So one way to do it is to just find nxn-k linearly independent, non-symmetric nxn matrices.

Another way to do it is to create a specific basis and express all nxn matrices as a linear combination of the vectors from the symmetric basis and this non-symmetric basis.

Answer 4

Another way to do it is to create a specific basis and express all nxn matrices as a linear combination of the MATRICES from the symmetric basis and this non-symmetric basis.

Answer 5

Thanks

Answer 6

No problem. Unfortunately, there was a mistake:

"So one way to do it is to just find nxn-k linearly independent, non-symmetric nxn matrices."

For example for 2x2 matrices

C1 and C2 and independent and non-symmetric but C1 + C2 = S3 so you don't need both for the complemetary base. But that should be a hint for the explicit construction if you haven't figured it out yet.