Proof of Theorem CNMB

(⇒) Suppose that the columns of $A$ are a basis for $\complex{m}$. Then Definition B says the set of columns is linearly independent. Theorem NMLIC then says that $A$ is nonsingular.

(⇐) Suppose that $A$ is nonsingular. Then by Theorem NMLIC this set of columns is linearly independent. Theorem CSNM says that for a nonsingular matrix, $\csp{A}=\complex{m}$. This is equivalent to saying that the columns of $A$ are a spanning set for the vector space $\complex{m}$. As a linearly independent spanning set, the columns of $A$ qualify as a basis for $\complex{m}$ (Definition B).