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Math Help - Finite Dimensional Vector Space

  1. #1
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    Finite Dimensional Vector Space

    Let V, W be finite dimensional vector spaces over a
    field k and let Z subset of W be a

    subspace. Let T : V -> W be a linear map. Prove that

    dim( T^(-1) ( Z ) ) <= dim V - dim W + dim Z
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  2. #2
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    Quote Originally Posted by ques View Post
    Let V, W be finite dimensional vector spaces over a
    field k and let Z subset of W be a

    subspace. Let T : V -> W be a linear map. Prove that

    dim( T^(-1) ( Z ) ) <= dim V - dim W + dim Z
    your inequality is not correct. it should be \geq instead of \leq. let T^{-1}(Z)=X and define the map S: V \longrightarrow W/Z by S(v)=T(v) + Z. clearly \ker S = X and thus V/X \cong S(V)=T(V)/Z.

    hence \dim V - \dim X = \dim V/X = \dim T(V)/Z = \dim T(V) - \dim Z \leq \dim W - \dim Z. therefore \dim X \geq \dim V - \dim W + \dim Z.
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