I understand why the elementary row operations do not change the row space, but why they do not change the column space (or at least the dependence between the columns)?
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the same reason why column operations do not change the row space ! – alkabary Jun 07 '15 at 20:15
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1They do affect the column space (perform some row ops on $\bigl[ {1\ 0 \atop 1\ 0 }\bigr]$). To see why dependence relations are unchanged, look at the answer here. – David Mitra Jun 07 '15 at 20:21
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Related. – A.P. Jun 07 '15 at 21:21