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Math Help - Linear Transformation/isomorphism question

  1. #1
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    Linear Transformation/isomorphism question

    Could someone please explain to me how to do the following question


    Let W be a proper subset of an vectior space V, and let T be the projection onto W.
    Prove that T is not an isomorphism
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  2. #2
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    Re: Linear Transformation/isomorphism question

    It really depends how much linear algebra you know. There are so many answers to this question depending on implicit knowledge.



    The simplest way is to do the following:

    Pick elements w\in W, v_1,v_2\in V-W with v_1\not= v_2. Then T(w,v_1)=T(w,v_2)=w, so T cannot be injective.


    This works for both finite and infinite dimensional vector spaces. If your vector space is finite, there are other ways of looking at this:

    For example, if T is projection of V onto a proper subset W, then the associated matrix of T is not square and cannot be invertible.

    Alternatively, if T is an isomorphism, there is a 1-1 correspondence between a basis of V and a basis of W. If your vector space V is finite dimensional, then you are looking for a 1-1 correspondence between two finite sets of different cardinality, but this is impossible.
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