Let F be a field of characteristic 0 and let
W = ( A = [] ∈ : tr(A) = = 0) .
For = 1,…, n with , let be the n×n matrix with (i, j)-th entry 1 and all the remaining entries 0. For =2,…,n let be the n×n matrix with (1, 1) entry −1, ()-th entry +1 , and all remaining entries 0. Let S = { : = 1,…, n and } ∪ { : i = 2,…, n} .
[Note: You can assume, without proof, that S is a linearly independent subset of .]
(1) Prove that W is a subspace of and that W = span(S) . What is the dimension of W ?
(2) Suppose that is a linear functional on such that
(a) f(AB) = f(BA) , for all A, B ∈ .
(b) f(I) = n , where I is the identity matrix in .
Prove that f(A) = tr(A) for all A ∈ .