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If I have a matrix $A \in R^{(m \times n)}$ with $m \leq n$. All rows in the matrix are linearly independent. Does it hold that I can select any $m$ columns from $A$ and they will also be linearly independent?

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No. Take

$$ A = \left( \begin{array}{ccc} 1 & 0 & 0 & 0& 0& 0&\cdots \\ 0 & 1 & 0 &0& 0& 0&\cdots\\ 0 & 0 & 1 &0& 0& 0&\cdots \end{array} \right) $$

Then "all rows in the matrix are linearly independent" and it you can't " select any $3$ columns from $A$ and they will also be linearly independent".

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Thanks, not a very smart question. –  entropy Jan 2 '13 at 12:38
    
I reformulated my question here math.stackexchange.com/questions/269144/… –  entropy Jan 2 '13 at 12:56
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