2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: : R +R a11 a12 a13 ain ... a21 a22 a23 a2n ... a32 Ha11 + a22 + a33 +...+ ann = > ai. a31 a33 ... i=1 an2 An3 ann ... anl Let In be the n x n identity matrix (or equivalently, a vector in R"). Calculate (A) that tr(In) = = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and brl. Then compute tr(AB) and tr(BA).

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