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Thread: find a column of the matrix that can be deleted

  1. #1
    Super Member bigwave's Avatar
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    find a column of the matrix that can be deleted

    $\textsf{Determine if the columns of the matrix span $R^4$.}\\$
    $\textit{Then, find a column of the matrix that can be deleted and yet have the remaining matrix columns still span $R^4$.}$
    $$\left[\begin{array}{rrrrr}
    12& -7& 11& -9 &5 \\
    -9& 4& -8& 7& -3 \\
    -6& 11& -7& 3&-9\\
    4&-6&10&-5&12
    \end{array}\right]$$

    ok we are supposed to solve this
    using SAGE

    I presume the first step is row reduction
    quidence requested☕
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  2. #2
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    Re: find a column of the matrix that can be deleted

    Part of your post seems to have been cut off. I see "find a column of the matrix that can be deleted and still have the remaining matrix". "Have the remaining matrix" what? Still span $\displaystyle R^4$?

    Yes, row reduce the matrix. I have no idea what "SAGE" is. Do you know how to row reduce a matrix?
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  3. #3
    MHF Contributor
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    Re: find a column of the matrix that can be deleted

    Quote Originally Posted by bigwave View Post
    $\textsf{Determine if the columns of the matrix span $R^4$.}\\$
    $\textit{Then, find a column of the matrix that can be deleted and yet have the remaining matrix columns still span $R^4$.}$
    $$\left[\begin{array}{rrrrr}
    12& -7& 11& -9 &5 \\
    -9& 4& -8& 7& -3 \\
    -6& 11& -7& 3&-9\\
    4&-6&10&-5&12
    \end{array}\right]$$

    ok we are supposed to solve this
    using SAGE

    I presume the first step is row reduction
    quidence requested☕
    SAGE looks pretty straightforward to use. I'll leave that bit to you.

    define a matrix m initialized as you've written it.

    call m.echelon_form(). The dimension of the span of m is 5-(# of rows of all 0's in the echelon form)

    next for each column, delete it from m (there must be some easy way to do this in SAGE) and find the determinant.

    If the determinant is non-zero then you know that the 4 remaining columns span $\mathbb{R}^4$

    I find that all but the last column may be removed and the remaining 4 columns are full rank.

    The first 4 columns are not linearly independent.
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  4. #4
    Super Member bigwave's Avatar
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    Re: find a column of the matrix that can be deleted

    ok I got this \\

    $\textit{The reduced echalon form is:}$
    $$\left[\begin{array}{rrrrr}\displaystyle
    1& 0& 0& \displaystyle\frac{-10}{21}& 0\\ \\
    0& 1& 0& \displaystyle\frac{-25}{84}& 0\\ \\
    0& 0& 1& \displaystyle\frac{-40}{84}& 0\\ \\
    0& 0& 0& 0& 1
    \end{array}\right]$$

    so assume the $R_4$ and $C_5$ can be removed?
    but we can just remove $C_5$ and still have $\mathbb{R^4}$ ... can't we

    Never tried SAGE before but we have 14 problems we are supposed to use with it.

    Ok I was expecting a reduced row calculator on SAGE but didn't find one .... maybe there is'

    here is the link to SAGE http://linear.ups.edu/html/sage
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  5. #5
    Super Member bigwave's Avatar
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    Re: find a column of the matrix that can be deleted

    do we just remove columns to ck $R^4$
    or will reduced rows better way
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