# Solving Linear Systems - Criteria for Rank Condition

It is well known that for $$A \in \mathbb{K}^{n \times m}$$ and $$b \in \mathbb{K}^n$$ the system $$Ax = b$$ is solvable if and only if $$\operatorname{rank}(A) = \operatorname{rank}(A \mid b)$$.

I suppose, in practice one would just use Gauß-Algorithm. However, is there any criterion for $$\operatorname{rank}(A) = \operatorname{rank}(A \mid b)$$?

$${\rm rank}( A ) = {\rm rank}( A | b )$$ if and only if $$b$$ is in the column space of $$A$$. Is that what you are looking for?