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Math Help - 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 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.
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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 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
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