Proof of Theorem CMVEI

Suppose that the augmented matrix of the system of equations is row-equivalent to $B$, a matrix in reduced row-echelon form with $r$ nonzero rows. Because $B$ has $m$ rows in total, the number of nonzero rows is less than or equal to $m$. In other words, $r\leq m$. Follow this with the hypothesis that $n>m$ and we find that the system has a solution set described by at least one free variable because \begin{equation*} n-r\geq n-m>0. \end{equation*}

A consistent system with free variables will have an infinite number of solutions, as given by Theorem CSRN.