Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F.
Tr(M_n) = sum of diagonal elements
The Attempt at a Solution
I view the trace Tr as a linear transformation Tr: M_n (F) -> F. I find the dimension of the subspace spanned by M_n (F) by using the fact:
dim M_n (F) = dim Tr [M_n (F)] + dim null [M_n (F)]
Since dim M_n (F) = n^2, we have
n^2 = dim Tr [M_n (F)] + dim null [M_n (F)]
I'm confused on what's in the right part of the above equation:
Isn't dim Tr [M_n (F)] = 0 since Tr [M_n (F)] = 0?
Shouldn't dim null [M_n (F)] =n^2 - dim Tr [M_n (F)] = n^2 - 1?
Also, I know that the subspace is generated by the following matrices: A, B, and [A,B] (the commutator of A and B).