Proof of Theorem DFS

If $A$ row-reduces to a matrix in reduced row-echelon form with $r$ nonzero rows, then the matrix $C$ of extended echelon form (Definition EEF) will be an $r\times n$ matrix in reduced row-echelon form with no zero rows and $r$ pivot columns (Theorem PEEF). Similarly, the matrix $L$ of extended echelon form (Definition EEF) will be an $m-r\times m$ matrix in reduced row-echelon form with no zero rows and $m-r$ pivot columns (Theorem PEEF).

\begin{align*} \dimension{\nsp{A}} &=\dimension{\nsp{C}}&&\knowl{./knowls/theorem.FS.knowl}{\text{Theorem FS}}\\ &=n-r&&\knowl{./knowls/theorem.BNS.knowl}{\text{Theorem BNS}}\\ \ \\ \dimension{\csp{A}} &=\dimension{\nsp{L}}&&\knowl{./knowls/theorem.FS.knowl}{\text{Theorem FS}}\\ &=m-(m-r)&&\knowl{./knowls/theorem.BNS.knowl}{\text{Theorem BNS}}\\ &=r\\ \ \\ \dimension{\rsp{A}} &=\dimension{\rsp{C}}&&\knowl{./knowls/theorem.FS.knowl}{\text{Theorem FS}}\\ &=r&&\knowl{./knowls/theorem.BRS.knowl}{\text{Theorem BRS}}\\ \ \\ \dimension{\lns{A}} &=\dimension{\rsp{L}}&&\knowl{./knowls/theorem.FS.knowl}{\text{Theorem FS}}\\ &=m-r&&\knowl{./knowls/theorem.BRS.knowl}{\text{Theorem BRS}}\\ \end{align*}