I would still like to characterize the solutions to a rank-defficient matrix equation. I know that this can be done using the rank-nullity theorem, but I'm having a hard time precisely characterize this relationship using basis vectors for the matrix's range and null space.
More formally, let's suppose I have a system of equation $Ax = b$, where $A\in\mathbb{R}^{n\times n}$ is rank-$m<n$ and $x,b\in\mathbb{R}^n$. Suppose that the range of $A$ is spanned by unit vectors $\{u_1,u_2, \dots, u_m\}$. By the rank-nullity theorem, its null space must be the span of vectors $\{v_1, v_2, \dots, v_{n-m}\}$. What is the general formula for solutions $x$, using the notation of these vectors?