# Math Help - Prove This Is A B-Invariant Subspace

1. ## Prove This Is A B-Invariant Subspace

Let V be a vector space over a field K. Let $A, B:V \rightarrow V$ be two linear maps. Given any polynomial $f \in K[t]$ let $V_f(A) = \ker f(A)$.

Prove that if AB=BA, then $V_f(A)$ is a B-invariant subspace of V.

2. Originally Posted by mathematicalbagpiper
Let V be a vector space over a field K. Let $A, B:V \rightarrow V$ be two linear maps. Given any polynomial $f \in K[t]$ let $V_f(A) = \ker f(A)$.

Prove that if AB=BA, then $V_f(A)$ is a B-invariant subspace of V.

$\forall x\in \ker f(A)\,,\,f(A)x=0\Longrightarrow$ $0=B(0)=Bf(A)x=f(A)Bx\Longrightarrow Bx\in\ker f(A)=V_f(A)$ and we're done.

The fact that $Bf(A)=f(A)B$ follows easily from $AB=BA$ and a little induction on $\deg f$.

Tonio