$\newcommand{rank}{\operatorname{rank}}\newcommand{nullity}{\operatorname{nullity}}$ Sorry if this question is too basic, but I'm wondering something. So the Fundamental Theorem of Linear Algebra says that if you have a $m \times n$ matrix $A$, then $n = \rank(A) + \nullity(A)$. But what does that really mean? For example, consider:
$ A = [1, 0, -1]$. $N(A) = [k, j, k]^T$ where k and j are any integer. I suppose I don't understand the dimension theorem holds up here. Is $\rank(A)$ equal to 1 (because of 1 row), or 3, because of 3 columns? A similar question for $N(A)$. Regardless, the sum of $\rank(A)$ and $\nullity(A)$ will not equal n in this case, so how exactly does the dimension's theorem hold up?
Furthermore, I wonder about a general case. Let's say you have a 4 x 4 matrix, and the null space is 4 dimensional. What exactly does that mean? I understand it to be that the null space has rank 4, so $\nullity(A)$ = 4, so by the dimension's theorem, $\rank(A)$ must be 0 right i.e. the zero matrix? I suppose I just don't see how it all intuitively fits together.
Thanks for your help.