[Math] Do row operations change the column space of a matrix

linear algebramatricesmatrix equationssystems of equations

I know that

(i) row operations do not change the row space

(ii) column operations do not change the column space

and (iii) row rank = column rank (but this is sort of unrelated, I think).

But, is it true that row operations do not change both the row space and the column space of a matrix?

Thanks,

EDIT: I am guessing that it's most likely true, since in Guassian elimination, solving Ax=b involves only row operations — there's something about column operations that makes the algorithm not work, I think (according to the book by Friedberg, Insel and Spence.)

Best Answer

Row operations in general do change the column space. Consider the following matrix

$$\left(\begin{array}{rrr} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 1 & 1 & -1\end{array}\right).$$

Row reducing, we get

$$\left(\begin{array}{rrr} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 0 & 0 & 0\end{array}\right).$$

The span of these columns is the set $\left\{\left(\begin{array}{c} a \\ b \\ 0\end{array}\right):a,b\in\mathbb{R}\right\}$ but this is not the span of the original columns.

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