Rank of matrix in determining linear dependence

**Craka** · March 27th 2009, 03:24 PM

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?

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

$ \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ \end{array}} \right) $

$ \left( {\begin{array}{*{20}c} 1 & 2 & 0 & 3 & 5 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right) $

**NonCommAlg** · March 27th 2009, 05:42 PM

Originally Posted by Craka

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?

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

$ \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ \end{array}} \right) $

$ \left( {\begin{array}{*{20}c} 1 & 2 & 0 & 3 & 5 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array}} \right) $

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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