Existence and uniqueness of solutions to $Ax=b$ in terms of number of equations and number of variables

Question

I roughly know that $\textbf Ax=b$ where $\textbf A\in\mathbb R ^{m\times n}$ has the following properties:

Existence: $rank(A)=m$ means a solution exists (because the column space would span all of $\mathbb R^m$), $rank(A)<m$ means a solution might not exist

Uniqueness: $rank(A)=n \implies nullity(A)=0 \implies$ any solution is unique, and $rank(A)<n \implies nullity(A)>0 \implies$ any solution, if it exists, is not unique (because since $Ay=0$ for some nonzero y, and if x is a solution to Ax=b, then A(x+y)=Ax+Ay=b+0=b shows that x+y is also a solution).

This works out and all, but is there a way to think about this in terms of the number of equations and number of variables in the system of equations? Like, for existence, rank(A)=m=the number of equations means a solution exists and for uniqueness, rank(A)=n=the number of variables means any solution is unique. I'm hoping there can be a more intuitive or simpler way to think about it.

Answer 1

There is this general result:

The linear system $Ax=b$ has solutions if & only if the matrix $A$ and the augmented matrix $[A|b]$ have the same rank. If it is the case, this common rank is the codimension of the (affine) subspace of solutions.