Find a basis for the space of 2x2 symmetric matrices

[ryanclarkeatm]

a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis.
b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.

 

[ebola1717]

The space of 2x2 matrices is in general isomorphic to a very familiar space. Think about the way addition of matrices and scalar multiplication work, and you should figure this out (and if you think about this for a while, you might realize a more general property about finite vector spaces over a field). From there, you should realize that the symmetric matrices are a subspace. If you look at some examples of 2x2 symmetric matrices, you should see the pattern.

 

[HallsofIvy]

A general 2 by 2 symmetric matrix is of the form
$$\begin{bmatrix}a & b \\ b & c \end{bmatrix}$$
you should be able to get a basis and the dimension from that.