Proof of Theorem CSCS

(⇒) Suppose $\vect{b}\in\csp{A}$. Then we can write $\vect{b}$ as some linear combination of the columns of $A$. By Theorem SLSLC we can use the scalars from this linear combination to form a solution to $\linearsystem{A}{\vect{b}}$, so this system is consistent.

(⇐) If $\linearsystem{A}{\vect{b}}$ is consistent, there is a solution that may be used with Theorem SLSLC to write $\vect{b}$ as a linear combination of the columns of $A$. This qualifies $\vect{b}$ for membership in $\csp{A}$.