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Math Help - Rank of matrix in determining linear dependence

  1. #1
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    Rank of matrix in determining linear dependence

    For the following three augmented matrices, would it be true to say that all of the the coefficient columns are linearly dependent, as the rank of the matrix in each case is less than the amount of coefficient columns?

    <br />
\left( {\begin{array}{*{20}c}<br />
   1 & { - 6} & 0 & 0 & 3 & { - 2}  \\<br />
   0 & 0 & 1 & 0 & 4 & 7  \\<br />
   0 & 0 & 0 & 1 & 5 & 8  \\<br />
   0 & 0 & 0 & 0 & 0 & 0  \\<br />
\end{array}} \right)<br />

    <br />
\left( {\begin{array}{*{20}c}<br />
   1 & 0 & 0 & 1 & 2  \\<br />
   0 & 1 & 0 & 2 & 1  \\<br />
   0 & 0 & 1 & 1 & 0  \\<br />
\end{array}} \right)<br />

    <br />
\left( {\begin{array}{*{20}c}<br />
   1 & 2 & 0 & 3 & 5  \\<br />
   0 & 0 & 1 & 2 & 3  \\<br />
   0 & 0 & 0 & 0 & 0  \\<br />
\end{array}} \right)<br />
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Craka View Post
    For the following three augmented matrices, would it be true to say that all of the the coefficient columns are linearly dependent, as the rank of the matrix in each case is less than the amount of coefficient columns?

    <br />
\left( {\begin{array}{*{20}c}<br />
1 & { - 6} & 0 & 0 & 3 & { - 2} \\<br />
0 & 0 & 1 & 0 & 4 & 7 \\<br />
0 & 0 & 0 & 1 & 5 & 8 \\<br />
0 & 0 & 0 & 0 & 0 & 0 \\<br />
\end{array}} \right)<br />

    <br />
\left( {\begin{array}{*{20}c}<br />
1 & 0 & 0 & 1 & 2 \\<br />
0 & 1 & 0 & 2 & 1 \\<br />
0 & 0 & 1 & 1 & 0 \\<br />
\end{array}} \right)<br />

    <br />
\left( {\begin{array}{*{20}c}<br />
1 & 2 & 0 & 3 & 5 \\<br />
0 & 0 & 1 & 2 & 3 \\<br />
0 & 0 & 0 & 0 & 0 \\<br />
\end{array}} \right)<br />
    in general in any m \times n matrix with entries in a field, if n > m, then the columns of the matrix are linearly dependent. the reason is that the the dimension of the column space (rank) is at most m,

    and we have n vectors (columns) in that space. since n > m, those vectors cannot be linearly independent.
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