I know that Row operations do not change the Row space of a matrix but how to know whether they have any effect on the column space or not? I tried to search but could not find a suitable explanation for this question. Please give some explanation, do elementary Row operations change the Column Space of a matrix?


closed as off-topic by GNUSupporter 8964民主女神 地下教會, Saad, Shaun, jvdhooft, Arturo Magidin Apr 7 '18 at 12:55

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  • $\begingroup$ Think about elementary row operations in terms of matrix operators. Are these elementary matrices of full rank? $\endgroup$ – jaslibra Apr 7 '18 at 12:30
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Yes of course, row operations preserves row space but in general not the colums space.

Let consider for example

$$\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right)\stackrel{R3-R2}\to\left(\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 0\end{array}\right)$$


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