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Math Help - Explicit Isomorphism (theoretical example)

  1. #1
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    Explicit Isomorphism (theoretical example)

    I haven't been having problems finding explicit isomorphisms and proving them for practical vector spaces, I am however baffled by this theoretical isomorphism question.

    QUESTION:
    Let V and W be vector spaces with dim V = n, dim W = m
    Let L : V -> W be a linear mapping
    Let A be the matrix L with respect to bases B for V, and C for W

    Define an explicit isomorphism from Range(L) to Col(A). Prove that your map is an isomorphism.

    ATTEMPT:
    Since A is the matrix L with respect to bases B and C, can I deduce that B and C are of the same size, and therefore, that V and W have the same dimension?

    This would mean that V and W are one-to-one iff they are onto, and if I can use this to show that V and W are onto, then Range(L) = W.

    But how can I show that V and W are one-to-one in order to do this, and what can I do with Col(A)?

    Any help would be greatly appreciated, thanks.
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  2. #2
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    Quote Originally Posted by crymorenoobs View Post
    I haven't been having problems finding explicit isomorphisms and proving them for practical vector spaces, I am however baffled by this theoretical isomorphism question.

    QUESTION:
    Let V and W be vector spaces with dim V = n, dim W = m
    Let L : V -> W be a linear mapping
    Let A be the matrix L with respect to bases B for V, and C for W

    Define an explicit isomorphism from Range(L) to Col(A). Prove that your map is an isomorphism.

    ATTEMPT:
    Since A is the matrix L with respect to bases B and C, can I deduce that B and C are of the same size, and therefore, that V and W have the same dimension?
    No, you can't! A will have n columns and m rows. It is not necessarily true that A is a square matrix whch is what you would be assuming.

    This would mean that V and W are one-to-one iff they are onto, and if I can use this to show that V and W are onto, then Range(L) = W.
    This makes no sense at all. V and W are vector spaces, not linear transformations. It makes no sense to say vector spaces are "one-to-one" or "onto". Take a deep breath and think about what you are saying. It is the linear transformation L that you are to show is "one-to-one" and "onto".

    But how can I show that V and W are one-to-one in order to do this, and what can I do with Col(A)?

    Any help would be greatly appreciated, thanks.
    When L is written as a matrix, "with respect to bases B for V, and C for W", you get the columns of A by applying L to each of the vectors in B in turn, writing the result as linear combination of vectors in C. That means, in particular, that if [tex]a_1, a_2, \cdot\cdot\cdot a_m[\math] is the first column, The L(v_1)= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n where v_1 is the first vector in basis B and w_1, w_2, \cdot\cdot\cdot w_n are the vectors in basis C. That is, in that sense, every column of A represents a vector in W.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    No, you can't! A will have n columns and m rows. It is not necessarily true that A is a square matrix whch is what you would be assuming.
    I realized this shortly after posting my question, Thanks.

    Quote Originally Posted by HallsofIvy View Post
    This makes no sense at all. V and W are vector spaces, not linear transformations. It makes no sense to say vector spaces are "one-to-one" or "onto". Take a deep breath and think about what you are saying. It is the linear transformation L that you are to show is "one-to-one" and "onto".
    I was just using terminology from my textbook, which said that if L : V -> W was an isomorphism, then V and W were said to be isomorphic. I thought it was implied in this context that they would be isomorphic with respect to L, the linear transformation.

    Quote Originally Posted by HallsofIvy View Post
    When L is written as a matrix, "with respect to bases B for V, and C for W", you get the columns of A by applying L to each of the vectors in B in turn, writing the result as linear combination of vectors in C. That means, in particular, that if [tex]a_1, a_2, \cdot\cdot\cdot a_m[\math] is the first column, The L(v_1)= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n where v_1 is the first vector in basis B and w_1, w_2, \cdot\cdot\cdot w_n are the vectors in basis C. That is, in that sense, every column of A represents a vector in W.
    Ok Thanks
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