please solve number 3 3.Let n = 4 (we choose 4 for calculation simplicity, but the followingare true for arbitrary n). Let S: R" →R be a linear transformation such that S(In) = n, andS(AB) = S(BA) for all n x n matrices A, B. In this problem, we will show that S and tr (definedin problem 2) are the same linear transformation.(A) Let i e {1, 2, - . , n}. Let ei € R"² be the matrix with 1 on the (i, i)-entry, and 0everywhere else. Calculate that S(e) = 1.Hint. Let j€ {1, 2, -.. ,n} with i # j. Calculate S(eij@ji) and S(ej;eij) respectively.Let i, j e {1,2, ·. ,n} with i # j. Let eij e R" be the matrix with 1 on the(B)(i, j)-entry, and 0 everywhere else. Calculate that S(eis) = 0.Hint. Calculate S(eijejj) and S(ejjeij) respectively.(C) Let A be anxn matrix such that tr(A) = 0. Use (1) and (2) to conclude S(A)= 0.(D) Use problem 1 to conclude S(X)tr(X), for all n x n matrix X.1 1.„ Let S, T : R"m → R be linear transformations. Suppose that ker S C ker T.Show that there exists a fixed scalar a E R such that T(x) = aS(x) for all x E R™.Hint. Use rank-nullity theorem.2.trace.Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n).Consider the following linear transformation:tr : Rn?a11a12a13aina21а22a23A2nna31a32a33a3nHaji + a22+ a33 + · · · + ann =i=1anl an2 an3Ann(1) .that tr(In) = n.Let In be the n x n identity matrix (or equivalently, a vector in Rn). Calculate(2)tr(AB) =Let A, B be two n x n matrices (or equivalently, a vector in Rn). Calculate thattr(BA).Hint. Write A = [aij] and B = [bki]. Write down the diagonal of AB and BA in terms of aijand bki. Then cотрute tr(AB) and tr(BA).

Name: please solve number 3 3.Let n = 4 (we choose 4 for calculation simplic
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: please solve number 3 3.Let n = 4 (we choose 4 for calculation simplicity, but the followingare true for arbitrary n). Let S: R" →R be a linear transformation such that S(In) = n, andS(AB) = S(BA) for all n x n matrices A, B. In this problem, we will