I need help to prove:
Given two matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times k}$. The following equivalence is true: $$ \mathcal{R}(A) \supseteq \mathcal{R}(B) \quad \Leftrightarrow \quad \mathrm{rank} A = \mathrm{rank} (A | B). $$ $\mathcal{R} (\cdot)$ denotes the image of a matrix and $(A | B)$ is a matrix with the columns of $A$ and $B$ next to each other.
I don't really know how to start with the proof. Could someone give me a hint?
$rank(A)$ is the dimension of $\mathcal{R}(A)$, which is the span of the columns of $A$. Since $\mathcal{R}(B)\subset\mathcal{R}(A)$, the columns of $B$ are in the span of the columns of $A$, so the dimension of the column space does not increase when you attach $B$ to $A$. Hence the rank does not change.
The reasoning can be reversed. The rank is unchanged as long as you add columns which are linear combination of the previous ones.