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: 
Sometimes you need to know all antiderivatives of a function, rather than just one antiderivative. For example, suppose you're solving the differential equation
$$
y'=-4y
$$
and there's an initial condition $y(0)=5$. You get
$$
\frac{dy}{dx}=-4y
$$
$$
\frac{dy}{y} = -4\;dx
$$
$$
\int\frac{dy}{y} = \int -4\;dx
$$
$$
\log\_e y = -4x + C.
$$
Here you've added a constant. Then
$$
y = e^{-4x+C} =e^{-4x}e^C
$$
$$
5=y(0)= e^{-4\cdot0}e^C = e^C
$$
So
$$
y=5e^{-4x}.
$$

(And sometimes you need only one antiderivative, not all of them. For example, in integration by parts, you may have $dv=\cos x\;dx$, and conclude that $v=\sin x$.)
The "integration constant" $C$ does have the "purpose" to make a seemingly true equation at least halfway true. In fact it is what many people call a "dangling variable", similar to the $i$ and $k$ when we talk about a "matrix $\bigl[a\_{ik}\bigr]\ $". As noted in other answers, an *indefinite integral* is not a function, but an infinite set of functions, any two of them differing by a constant. Writing $\langle F(x)\rangle$ (or similar) instead of $F(x)+C$ for the set of all functions differing from the term $F(x)$ by a constant, one could write, e.g.,
$$\int (5x -6x^4)\ dx =\langle {5\over2} x^2-{6\over5} x^5\rangle\ ,$$
and the mysterious constant has disappeared.