I have two definitions of null space. One by Serge Lang
Suppose that for every element $u$ of $V$ we have $\langle u,u\rangle=O$. The scalar product is then said to be null, and $V$ is called a null space.
and another by David C. Lay
The null space of an $m \times n$ matrix $A$ is the set of all solutions of the homogeneous equation $Ax=0$.
I cannot find the relationship between these definitions. Can anybody give a hint to me?
The two things described are very different; the only "relationship" is that the term null refers to something having to do with zero, and both things are some sort of vector space.
Those who use Lang's definition of a null-space would strictly refer to the second idea (what Lay calls a "null space") as the "kernel" of a matrix.