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Thread: Prove This Is A B-Invariant Subspace

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

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

    Prove that if AB=BA, then $\displaystyle V_f(A)$ is a B-invariant subspace of V.
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  2. #2
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let V be a vector space over a field K. Let $\displaystyle A, B:V \rightarrow V$ be two linear maps. Given any polynomial $\displaystyle f \in K[t]$ let $\displaystyle V_f(A) = \ker f(A)$.

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

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

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

    Tonio
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