how to prove this subspace and matrix equality

Question

$A\in M_{n\times n}(\mathbb{R})$ and $B\in M_{n\times m}(\mathbb{R})$, rank(B)$=m\le n$

could anyone help me in How to show : $\text{im }[B, AB]= \text{ im }(B)+ A \text{im }(B)$

$[B,AB]$ is a $n\times 2m$ matrix right?

if $x\in \text{ im }[B, AB]\Rightarrow\exists y\in\mathbb{R}^{2m}\ni [B,AB]y=x\in\mathbb{R}^n $

this means $[By,ABy]\in\mathbb{R}^n\Rightarrow By\in\mathbb{R}^m \text{ and } ABy\in \mathbb{R}^{n-m}$

I am not able to proceed further, please help

Answer 1

Write a $2m$ dimensional vector as $[\frac{v}{u}]$ where $v,u$ are $m$-dimensional vectors. Then $$[B|AB][\frac{v}{u}]=Bv+ABu$$ Going over all the possible $u,v\in \mathbb{R}^m$ you get that $Im([B|AB])=Im(B)+AIm(B)$.