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Math Help - vector spaces and linear maps

  1. #1
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    vector spaces and linear maps

    Suppose that V and W are vector spaces. Let Hom (V, W ) be the set of all
    linear maps φ : V → W . Add and scale linear maps in the obvious way to
    make Hom (V, W ) into a vector space. Define a linear map

    and prove that your map is an isomorphism.

    Hom (Rp , Rq ) → Rpq ,

    HELP PLEASE!!!!!!!!!
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  2. #2
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    Re: vector spaces and linear maps

    You know, I presume, that, given bases for V and W, any linear map from V to W can be written as a matrix. So this question is asking you to construct a linear map from the set of p by q matrices (which each have pq numbers, of course) to the set of ordered "pq-tuples". There is an obvious way to do that: for example, mapping \begin{bmatrix}a & b & c \\ d & e & f\end{bmatrix} to \begin{pmatrix}a & b & c & d & e & f\end{pmatrix}. Try it.
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  3. #3
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    Re: vector spaces and linear maps

    specifically, since we know that \mathbb{R}^p,\mathbb{R}^q are vector spaces, for any two bases \alpha,\ \beta we can form, for any T \in \mathrm{Hom}(\mathbb{R}^p,\mathbb{R}^q), the matrix [T]_\beta^\alpha. explicitly, if \alpha = \{u_1,\dots,u_p\}, and \beta = \{v_1,\dots,v_q\}, then the columns of [T]_\beta^\alpha are:

    [T([v_j]_\alpha)]_\beta

    \alpha,\ \beta are usually (but not always) taken to be the standard bases, in which case we can drop the brackets and simply write: T(e_j) for the j-th column of [T].

    so the desired isomorphism is: \varphi:T \to [T]_\beta^\alpha

    note that there is no special property of \mathbb{R} used above, and we have exactly the same argument for any two vector spaces U,V with dim(U) = p, and dim(V) = q, over an arbitrary field F, that is:

    \mathrm{Hom}_F(U,V) \cong \mathrm{Mat}_F(\mathrm{dim}(V),\mathrm{dim}(U)) \cong F^{(\mathrm{dim}(U))(\mathrm{dim}(V))}

    where the second isomorphism is just the "vectorization of a matrix", given by concatenating the columns.
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