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Thread: 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 $\displaystyle \begin{bmatrix}a & b & c \\ d & e & f\end{bmatrix}$ to $\displaystyle \begin{pmatrix}a & b & c & d & e & f\end{pmatrix}$. Try it.
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  3. #3
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    Re: vector spaces and linear maps

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

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

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

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

    note that there is no special property of $\displaystyle \mathbb{R}$ 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:

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

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