For any matrices $A$ and $B$ of the same size, show that $\mathrm{Im}(A ,B) = \mathrm{Im}(A) + \mathrm{Im}(B)$.

Question

I just want to be more familiar with block matrices and while I am reading Fuhzen Zhang book, I found this problem

$\mathrm{Im}( A,B)$ means the range of the block matrix $$\begin{bmatrix} A & B\end{bmatrix}$$

i consider it as this

$$T=\begin{bmatrix} A & B\end{bmatrix}$$ $\mathrm{Im}(T)$={$\mathrm{Tx: x∈C^n}$} $$Tx=\begin{bmatrix}A & B\end{bmatrix} \begin{bmatrix}x1 &x2\end{bmatrix}^t$$$ $$Tx=Ax1 + Bx2$

so $\mathrm{Im}(T)$=$\mathrm{Im}(A)$+$\mathrm{Im}(B)$ is this true or i missed something and sorry if i have a code mistakes i am just new in here

Answer 1

$$T=\begin{bmatrix} A & B\end{bmatrix}$$ $\mathrm{Im}(T)$={$\mathrm{Tx: x∈C^n}$} $$Tx=\begin{bmatrix}A & B\end{bmatrix} \begin{bmatrix}x1 &x2\end{bmatrix}^t$$$ $$Tx=Ax1 + Bx2$

so $\mathrm{Im}(T)$=$\mathrm{Im}(A)$+$\mathrm{Im}(B)$