Do $(0,1)$ and $(1,0)$ span the same column space?

Question

Do $(0,1)$ and $(1,0)$ span the same column space? I have found a question that asks whether matrix B is a multiple of matrix A if both of them have the same 4 fundamental subspaces. The answer was that 2 invertible matrices can have the same 4 fundamental matrices and not be multiple of each others. An example provided: $\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}, \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$. However, as I asked in the beginning how can it be possible that $(1, 0)$ and $(0, 1)$ span the same column space?

Answer 1

The column space of each of the two matrices $$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}, \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$$

is a subspace generated by the columns of the matrix.

Since the columns of each matrix are two linearly independent vectors, the column space of each is a two dimensional subspace of $ R^2$ which is $ R^2$.

Thus they have the same column space.