- #1

- 36

- 0

## Homework Statement

Suppose S,T ∈ L(V) and S is invertible. Prove that if p ∈ P(F) is a polynomial, then p(S*T*S

^{-1})=S*p(T)*S

^{-1}.

## Homework Equations

none

## The Attempt at a Solution

Suppose by contradiction that for any p ∈ P(F),

p(S*T*S

^{-1})≠S*p(T)*S

^{-1}for any p ∈ P(F).

Since this is true for any p∈ P(F), let p=1x ∈ P(F). Then

1*(S*T*S

^{-1})≠S*(1)*(T)*S

^{-1}

This implies that T≠T, a contradiction. Therefore if p ∈ P(F) is a polynomial, then p(S*T*S

^{-1})=S*p(T)*S

^{-1}.

Last edited: