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If you have an m x n matrix and the rows are linearly independent, are the columns linearly independent too? I know that it's the case for n x n matrices. I've yet tried to think about it, and I think the answer is yes, for example the matrix ((1,2,3),(a,2a,3c)) has linearly dependent rows and columns but I'm not sure that it works for every m x n matrix

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  • $\begingroup$ Say you have $m$ rows and $n$ columns with $n>m.$ You can never have $n$ linearly independent $m$-dimensional vectors. $\endgroup$ – saulspatz Jun 8 at 13:34
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Take

$\begin{pmatrix}1 &1 &0 \\2 &0 &1\end{pmatrix}$

then the rows are linear independent, but the columns aren't, since for example for the first column you have$$\begin{pmatrix} 1 \\ 2\end{pmatrix}=\begin{pmatrix}1 \\0\end{pmatrix}+2\begin{pmatrix}0 \\ 1\end{pmatrix}$$

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  • $\begingroup$ thanks a lot ;) $\endgroup$ – Mari3 Jun 8 at 13:29
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  • Your set of vectors that form the matrix are linearly independent, iff your matrix is invertible.

  • only square matrices are invertible.

  • Therefore, only square matrices are linearly Independent.

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In $m\times n$ matrix, the maximum number of independent rows or columns possible is the order of the largest square you can get from it. If $m >n$ then order of the largest square is n, so you can get at most n linearly independent rows or columns (and vice versa). If you get x linearly independent rows then you will also get same number of linearly independent columns also (and vice versa), even for a rectangular matrix.

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If you have an m x n matrix and the rows are linearly independent, are the columns linearly independent too?

This is only necessarily true when the matrix is square.

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