Question: Given two matrices $A\_{m\times n}$ and $B\_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?

I know $r(AB)=r(B)-\dim N(A) \cap R(B)$, so is it true the above iff $\dim N(A)\cap R(B)=0$? But seems incorrect if $m\lt p$. Please help.

Answer: 
The indefinite integral $\int f(x)dx$ is the function $F(x)$ such that $\frac{d}{dx}F(x) = f(x)$. Problem is, there are multiple (in fact infinitely many) such functions $F(x)$. We cannot allow the expression $\int f(x)dx$ to refer to multiple functions, so to get around this we introduce the *constant of integration* $C$. This fixes things, because though there are infinitely many $F(x)$ such that $\frac{d}{dx}F(x) = f(x)$, if we pick any single such function $F(x)$ then all solutions to $\frac{d}{dx}G(x) = f(x)$ are of the form $G(x) = F(x) + C$ for some value of $C$, while for any value of $C$ the function $F(x)+C$ satisfies $\frac{d}{dx}(F(x)+C) = f(x)$. Hence we say $\int f(x)dx = F(x) + C$.
Say every month you earned 10 dollars on a principal of 1000 dollars and put it aside. What have you got after 10 months ?

Not always 100 dollars or 180 dollars or 130. That is so, only if the box contained nothing not at start.

If it contained 35 dollars at start, that is the constant added to the accumulated (integrated) amount $A:$

$ A = m.10 + c = m.10 + 35 $ dollars or 135 or 235 dollars..

The same applies to more complicated integrals. It is also the *initial value* in integration of a variable function with initial value prescribed to evaluate boundary value differential equations.