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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
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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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  6. #6
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    Re: find a column of the matrix that can be deleted

    The problem specifically asks which column can be removed. It says nothing about rows!

    However, what you show is NOT "reduced echelon form". And if the reduced echelon form does NOT have a column that is all zeros, NO column can be removed.
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