Do row operations change the column space of a matrix?

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.