You said
> Isn't dim Tr [M_n (F)] = 0 since Tr [M_n (F)] = 0?
No, dim Tr [M_n (F)] = 1 since Tr [M_n (F)] = F
You have confused M_n (F) (all nxn matrices over F) with its subset of trace zero matrices. And a (non degenerate) field over itself obviously has dimension 1 because it's spanned by...