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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)?

user118837
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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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    They *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](http://math.stackexchange.com/questions/1144313/elementary-row-operation-on-matrices-and-column-space). – David Mitra Jun 07 '15 at 20:21
  • [Related](https://math.stackexchange.com/questions/332908/looking-for-an-intuitive-explanation-why-the-row-rank-is-equal-to-the-column-ran). – A.P. Jun 07 '15 at 21:21

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