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Math Help - Relations between transformaion and vector in/dependence

  1. #1
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    Relations between transformaion and vector in/dependence

    So we have vector space V over the field F, and we have transformation T:V->V
    (v1,v2,......,vn is in V)

    Claim: If Tv1,.......,Tv2 is linearly independent then v1,v2,......,vn is linearly independent.

    The claim is true (I think) but I can only explain it in words so basically it's not a proof. Can someone give me a hint how to prove this? Do I have to use the conditions of linear transformation (that it preserve addition and multiplication with scalar) and somehow get to the conclusion that the vectors are independent too? Thanks.
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  2. #2
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    Re: Relations between transformaion and vector in/dependence

    Suppose for contradiction that the input vectors v_n are linearly dependent. Then let V be the matrix made up of these v_n's.

    As the vectors are dependent V has a nonzero null space.
    So for any x in that null space Vx = 0.
    TVx = 0
    so the null space of TV is at least the null space of V. It could be higher dimensional if T isn't invertible.
    Thus the vectors of TV are not linearly independent.
    So the vectors v_n must be linearly independent.
    Thanks from davidciprut
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  3. #3
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    Re: Relations between transformaion and vector in/dependence

    Hi,
    Here's a direct proof that uses the definition of linear independence:

    Relations between transformaion and vector in/dependence-mhflinearalgebra9.png
    Thanks from Hartlw and davidciprut
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  4. #4
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    Re: Relations between transformaion and vector in/dependence

    I actually found the proof before reading the posts, but thank you guys a lot
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