16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.5.15

Linear Transformations

Mathematics — Linear Algebra · trace-of-left-multiplication-operator

Problem

According to Exercise 3, similar matrices have the same trace. Thus we can define the trace of a linear operator on a finite-dimensional space to be the trace of any matrix which represents the operator in an ordered basis. This is well-defined since all such representing matrices for one operator are similar. Now let

V

be the space of all

2 \times 2

matrices over the field

F

and let

P

be a fixed

2 \times 2

matrix. Let

T

be the linear operator on

V

defined by

T(A) = PA

. Prove that

\text{trace}(T) = 2\,\text{trace}(P)

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