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!

If your matrix is A (doesn't have to be square, can be nxm) and has rank $r< min(n,m)$, then the null space is spanned by ${max(n,m)-r}$ orthogonal vectors and is the space orthogonal to the $span$(A) $\in \mathbb{R}^{max(n,m)}$ (i.e., the linear combination of the basis/orthogonal vectors $\in \mathbb{R}^{max(n,m)}$ that are orthogonal to the $r$ basis vectors of A). See rank-nullity theorem.   In the simplest example, if $A=\left[\begin{array}{cc}
  1&0\\
  0&0
  \end{array}\right]
$, then $span(A)=\alpha\left[\begin{array}{c}
  1\\
  0
  \end{array}\right], \alpha \in \mathbb{R}$ and $null(A)=\beta\left[\begin{array}{c}
  0\\
  1
  \end{array}\right], \beta \in \mathbb{R}$
Play around with "null" in base Matlab, or SVD in Python like in this answer, where it can be seen that any zero eigenvalues correspond to eigenvectors in the left matrix that span the null space (also see here)
