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Math Help - Determining Linear Independence

  1. #1
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    Determining Linear Independence

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    Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

    Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

    [v1 | v2 | ... | vm]A




    What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
    1. Vectors v1, v2, ..., vm are linearly independent
    2. None of the vectors v1, v2, ..., vm is redundant
    3. ker[v1 | v2 | ... | vm] = {0}
    4. rank[v1 | v2 | ... | vm] = m


    Can anyone give me pointers in what to be looking at in order to answer this question?

    Thanks!
    Last edited by Oijl; February 9th 2011 at 06:08 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Oijl View Post
    Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

    Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

    [v1 | v2 | ... | vm]A




    What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
    1. Vectors v1, v2, ..., vm are linearly independent
    2. None of the vectors v1, v2, ..., vm is redundant
    3. ker[v1 | v2 | ... | vm] = {0}
    4. rank[v1 | v2 | ... | vm] = m


    Can anyone give me pointers in what to be looking at in order to answer this question?

    Thanks!
    Suppose that \displaystyle v_1=\sum_{k=2}^{m}\alpha_k v_k for example. Then, a pretty quick check shows that \displaystyle A(e_1)=A\left(\sum_{k=2}^{m}\alpha_k e_k\right) where \{e_1,\cdots,e_m\} is the canonical basis for \mathbb{R}^m but by injectivity this impiles that \displaystyle e_1=\sum_{k=2}^{m}\alpha_k e_k which is a contradiction.
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  3. #3
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    Quote Originally Posted by Oijl View Post
    Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

    Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

    [v1 | v2 | ... | vm]A




    What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
    1. Vectors v1, v2, ..., vm are linearly independent
    2. None of the vectors v1, v2, ..., vm is redundant
    3. ker[v1 | v2 | ... | vm] = {0}
    4. rank[v1 | v2 | ... | vm] = m


    Can anyone give me pointers in what to be looking at in order to answer this question?

    Thanks!
    Is m < n, m > n, or m = n?

    If that isn't giving, you can look at each case.
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  4. #4
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    Thanks for trying to help, but I got it now.
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