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Linear Algebra

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Please help with the following problem. Also, please be detailed so that I can understand how the problems were solved.

Let A = ( ) be an n n matrix. Define a trace of A to be the sum of the diagonal elements, that is
tr(A) = .

(a) Show that the trace is a linear map of the space of n n matrices into K.
(b) If B is invertible, show that tr(B AB) = tr(A).
(c) Prove that there are no matrices A, B such that AB-BA=

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Solution Summary

This solution is comprised of a detailed explanation to show that the trace is a linear map of the space of n n matrices into K.

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