Please solve number 3!, 1 and 2 are there for supporting information but don't need to be solved. 1.Let S, T: Rm → 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 Rm.Hint. Use rank-nullity theorem.trace.2.Let n =4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n).Consider the following linear transformation:2tr : R" →Ra12a13aina21A22а23a2nаз1a32аззa3nHai1 + a22 + a33 ++ anni=1An1An2An3AnnLet In be the n x n identity matrix (or equivalently, a vector in R"). Calculate(1) .that tr(In)= n.Let A, B be two n x n matrices (or equivalently, a vector in R"'). Calculate that(2)tr(AB) = tr(BA).Hint. Write A = [aij] and B = [bri]. Write down the diagonal of AB and BA in terms of aijand brl. Then compute tr(AB) and tr(BA). 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 E Rn² be the matrix with 1 on the (i, i)-entry, and 0everywhere else. Calculate that S(ei) = 1.Hint. Let je {1,2, ... ,n} with i + j. Calculate S(eijeji) and S(ejieij) respectively.(B) Let i, j e {1,2, -.. , n} with i + j. Let eij e R"² be the matrix with 1 on the(i, j)-entry, and 0 everywhere else. Calculate that S(eij) = 0.Hint. Calculate S(eijejj) and S(ejjeij) respectively.(C) Let A be a nxn 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.

Name: Please solve number 3!, 1 and 2 are there for supporting information b
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: Please solve number 3!, 1 and 2 are there for supporting information but don't need to be solved. 1.Let S, T: Rm → 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