Let $A$ be an $m \times n$ matrix and let $B$ be an $n \times p$ matrix. Show that $\text{rank } A + \text{nullity } A = n$ is the special case of $\text{rank } AB + \dim(\text{null } A \cap \text{col } B)=\text{rank } B$.
We know the product $AB$ is an $m \times p$ matrix and the rank of a matrix is the number of linearly independent columns. We also know that the nullity is the number of columns that contain all zeros. We also notice that the column space of $B$ and the null space of $A$ are respectively, "The span of $B$'s column vectors" and "set of all vectors that are mapped to the zero vector by matrix $A$." And many of us know this theorem, "$\text{rank } A + \text{nullity } A = n$". And from what I know, $\dim(\text{null } A \cap \text{col } B)$ is the dimension of the intersection of the two definitions I said about $\text{null } A$ and $\text{col } B$.