solve number 3!, 1 and 2 are there for supporting information but don’t need to be solved. Transcribed Image Text: 1. Let S, T: Rm

solve number 3!, 1 and 2 are there for supporting information but don’t need to be solved. Transcribed Image Text: 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:
2
tr : R” →R
a12
a13
ain
a21
A22
а23
a2n
аз1
a32
азз
a3n
Hai1 + a22 + a33 +
+ ann
i=1
An1
An2
An3
Ann
Let 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 aij
and brl. Then compute tr(AB) and tr(BA). Transcribed Image Text: 3.
Let n = 4 (we choose 4 for calculation simplicity, but the following
are true for arbitrary n). Let S: R” → R be a linear transformation such that S(In) = n, and
S(AB) = S(BA) for all n x n matrices A, B. In this problem, we will show that S and tr (defined
in 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 0
everywhere 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.

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