Does changing rows in matrix changes column space order?

Question

For example in matrix:

$\begin{bmatrix} 0 & -2 & 3\\ 4 & 0 & 11\end{bmatrix}$

Column Spaces are asked.

The answer is
{(4,0), (0,-2)}

Shouldn't we take the original matrix's columns which should be
{(0,4), (-2,0)}

Why did we change the order? Or the books answer is wrong?

Answer 1

While in this particular case, both the sets $\{(0,4),(-2,0)\}$ and $\{(4,0),(0,-2)\}$ span $\Bbb R^2$, which is the column space, in general, you cannot change the top-to-bottom order of the column vectors. For example, take $A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$. The column space is the span of $(1,0)$, not $(0,1).$