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?
If A is your matrix, the null-space is simply put, the set of all vectors v such that A \cdot v = 0. It’s good to think of the matrix as a linear transformation; if you let h(v) = A \cdot v, then the null-space is again the set of all vectors that are sent to the zero vector by h. Think of this as the set of vectors that lose their identity as h is applied to them.
Note that the null-space is equivalently the set of solutions to the homogeneous equation A \cdot v = 0.
Nullity is the complement to the rank of a matrix. They are both really important; here is a similar question on the rank of a matrix, you can find some nice answers why there.