The Wikipedia article on matrix rank has a properties section that includes the following given that $A$ is $m×n$:
- If $B$ is an $n×k$ matrix of rank $n$, then $rank(AB)=rank(A)$
- If $C$ is an $l×m$ matrix of rank $m$, then $rank(CA)=rank(A)$
I have not seen these results before, and actually was not able to find them mentioned elsewhere after quite a bit of searching, and Wikipedia does not provide or cite proofs. I am wondering what they are based on?
Here's an informal explanation that should offer some intuition.
We're given that $A$ is a transformation from $n$-dimensional space to some $\mathrm{rank} (A)$-dimensional part of $m$-dimensional space. As a visualization, picture $A$ being some transformation whose image is plane in $\Bbb R^3$.
We're then told that $B$ is a transformation from a $k$-dimensional space to the $n$-dimensional space and that it has an image of all of $n$-dimensional space (i.e. $\mathrm{rank} B = n$). The key observation to make here is that every vector that can be input into $A$ can be produced by $B$.
From this, we can see that the composition $AB$ will take vectors from some $k$-dimensional space, map them to the whole of the $n$-dimensional space, and then finally place them in the $\operatorname{rank}(A)$-dimensional subspace of the $m$-dimensional space that $A$ has an an image.
A similar line of reasoning can be used to visualize the meaning of the second statement.