The number of solutions to $Ax=b$, is either zero, or if a solution exists, then equal to the size of the null space of $A$

Question

The number of solutions to $Ax=b$, is either zero, or if a solution exists, then equal to the size of the null space of $A$

How can I prove this?

If the system was $Ax=0$ then it is clear to me as it directly follows from the definition. What about $b\neq 0$

Thanks in advance.

Answer 1

This is a basic fact, that follows from the result that $\{x|Ax=b\}= \{x_0+y| y\in\rm{ker} A\}$, where $x_0$ is any particular solution to $Ax=b$.

The proof is straight forward.