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!

Its the solution space of the matrix equation $AX=0$ . It includes the trivial solution vector $0$. If $A$ is row equivalent to the identity matrix, then the zero vector is the only element of the solution space. If it is not i.e. when the column space of $A$ is of a dimension less than the number of columns of A then the equation $AX=0$ has non trivial solutions which form a vector space, whose dimension is termed nullity.
