Do row operations change the column space of a matrix?

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.)

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.

The answer is no. I'll give an example that can be generalized to any matrix size and any field.

Consider $A=\begin{bmatrix} 1& 0\\1& 0\end{bmatrix}$. Applying row operation to annihilate the second row, we get $B=\begin{bmatrix} 1&0\\0& 0\end{bmatrix}$. The column space of $A$ is the subspace of all multiples of $[1\ 1]^T$. The column space of $B$ is the subspace of all multiples of $[1\ 0]^T$. Clearly, we can generalize this to any size by appending columns and rows of zeros to $A$ and $B$.

Note: Row operations do leave column dependencies unchanged. That is, if some linear combination of the columns of the original matrix is the zero-vector, then so is the same linear combinations of changed columns under row operations. However, as noted in other answers, the column space does not (in general) stay the same.

No it is not true.

Row operations leaves the row space and null space unchanged, but can change the column space.

That is, row operations do not affect the linear dependence relations among the columns, but can change the linear dependence relations among the rows.