Proof of Theorem RMRT

Suppose we row-reduce $A$ to the matrix $B$ in reduced row-echelon form, and $B$ has $r$ nonzero rows. The quantity $r$ tells us three things about $B$: the number of leading 1's, the number of nonzero rows and the number of pivot columns. For this proof we will be interested in the latter two.

Theorem BRS and Theorem BCS each has a conclusion that provides a basis, for the row space and the column space, respectively. In each case, these bases contain $r$ vectors. This observation makes the following go.

\begin{align*} \rank{A} &=\dimension{\csp{A}}&&\knowl{./knowls/definition.ROM.knowl}{\text{Definition ROM}}\\ &=r&&\knowl{./knowls/theorem.BCS.knowl}{\text{Theorem BCS}}\\ &=\dimension{\rsp{A}}&&\knowl{./knowls/theorem.BRS.knowl}{\text{Theorem BRS}}\\ &=\dimension{\csp{\transpose{A}}}&&\knowl{./knowls/theorem.CSRST.knowl}{\text{Theorem CSRST}}\\ &=\rank{\transpose{A}}&&\knowl{./knowls/definition.ROM.knowl}{\text{Definition ROM}} \end{align*}

Jacob Linenthal helped with this proof.