Question: Let A = [a; ] E F be a square n X n matrix. We define the trace of A to be the number Tr(A)

Let A = [a; ] E F"" be a square n X n matrix. We define the trace of A to be the number Tr(A) = akk . k=1 a) Prove that Tr : Fixn - F is a linear function. b) Let A, B E FX . Prove that Tr(AB) = Tr(BA). c) Show that the subspace of Fx" spanned by the matrices of the form AB - BA has dimension n2 - 1. d) If : Fox - F is a linear function such that (AB) = 4(BA) for all matrices A, B in Fx", prove that is a scalar multiple of the trace function Tr. e) Prove that Tr(PAP-) = Tr(A) for any invertible matrix P in F"*" . Explain carefully how this implies that the trace is independent of the basis chosen to express a given linear operator Te L(V, V) as a matrix, implying that we have a well-defined trace map Tr : L(V , V) - F for any vector space V over F

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