Kunze Linear Algebra · Problem 3.4.9

Linear Transformations

Mathematics — Methods · similar-matrices-same-operator

Problem

Let

V

be a finite-dimensional vector space over the field

F

and let

S

and

T

be linear operators on

V

. We ask: When do there exist ordered bases

\mathfrak{B}

and

\mathfrak{B}'

for

V

such that

[S]_{\mathfrak{B}} = [T]_{\mathfrak{B}'}

? Prove that such bases exist if and only if there is an invertible linear operator

U

V

such that

T = USU^{-1}

. (Outline of proof: If

[S]_{\mathfrak{B}} = [T]_{\mathfrak{B}'}

, let

U

be the operator which carries

\mathfrak{B}

onto

\mathfrak{B}'

and show that

S = UTU^{-1}

. Conversely, if

T = USU^{-1}

for some invertible

U

, let

\mathfrak{B}

be any ordered basis for

V

and let

\mathfrak{B}'

be its image under

U

. Then show that

[S]_{\mathfrak{B}} = [T]_{\mathfrak{B}'}