Physical meaning of the null space of a matrix What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank $r$ of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it tells me that, if I combined the vectors in its columns in some order, I'd get a set of coordinates for an $r$-dimensional space, where $r$ is minimum (please correct me if I'm wrong). So that means I can relate rank (and also dimension) to actual coordinate systems, and so it makes sense to me. But I can't think of any physical meaning for a null space... could someone explain what its meaning would be, for example, in a coordinate system?
Thanks!

In mechanical engineering, the example of $AX=B$ is found in the finite element method, where $KU=F$, in which $K$ is the stiffness matrix, $U$ is the displacement of nodes and $F$ represents lumped forces at the same nodes. While $KU=0$ as the $U$ isn't equal to the zero vector, it means that a system like a structure can move without any forces developed in, or it won't prevent displacement defined by $U$.  
