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Math Help - T-invariant subspace

  1. #1
    jax
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    T-invariant subspace

    Let V be an n-dimensional vector space over F (can be taken equal to R or C). Let a be a non-zero scalar and let T be the operator defined on a basis v1,...,vn of V by
    T(v1)=av1+v2, T(v2)=av2+v3,......T(vn-1)=avn-1+vn, T(vn)=avn.
    Prove that if W is a T-invariant subspace of V, and v1 belongs to W, then W=V.

    This is what I have so far:
    Suppose W is a T-invariant subspace. If v1 beongs to W, the T(v1)=T(W). Since v2 belongs to W, then T(v2)=T(W) implies T(W)=av2+v3, thus v3 belongs to W. Since v2, v3 belong to W then T(vn-1)=T(W) implies T(W)=avn-1+vn, thus vn belongs to W. Since vn beloings to W, then T(vn)=T(W) implies T(W)=avn, thus W=V.
    How does that look?....any suggestions? Thank you!!!
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  2. #2
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    Quote Originally Posted by jax View Post
    Let V be an n-dimensional vector space over F (can be taken equal to R or C). Let a be a non-zero scalar and let T be the operator defined on a basis v1,...,vn of V by
    T(v1)=av1+v2, T(v2)=av2+v3,......T(vn-1)=avn-1+vn, T(vn)=avn.
    Prove that if W is a T-invariant subspace of V, and v1 belongs to W, then W=V.

    This is what I have so far:
    Suppose W is a T-invariant subspace. If v1 beongs to W, the T(v1)=T(W).

    No. If W \,\,is \,\,T- invariant and v_1\in W\,\,then\,\,T(v_1)\in W


    Since v2 belongs to W,

    Why? This is true, but some explanation is required. This isn't trivial nor immediate.


    then T(v2)=T(W) implies T(W)=av2+v3, thus v3 belongs to W. Since v2, v3 belong to W then T(vn-1)=T(W) implies T(W)=avn-1+vn, thus vn belongs to W. Since vn beloings to W, then T(vn)=T(W) implies T(W)=avn, thus W=V.


    How "thus"?? What you wrote T(W)=av_n makes no sense... We have W=V because as above we've proven that W contains a basis of V. Period.

    Tonio


    How does that look?....any suggestions? Thank you!!!
    .
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