91Ó°ÊÓ

Assume that the system has matrix \(A\) of the size \(m \times n\). Then the dimension of the rows in \(A\) is equal to the rank of the matrix. So, \(\dim \,{\rm{Row}}\,A = {\rm{rank}}\,A\).

Also, as matrix \({A^T}\) has the size \(n \times m\), the dimension of columns in \(A\) is equal to the rank of matrix \({A^T}\). So, \(\dim \,{\rm{Col}}\,A = {\rm{rank}}\,{A^T}\).

\(\begin{array}{c}{\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\\\dim \,{\rm{Row}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n.\end{array}\)

b. By the dual of the rank theorem, \({\rm{rank}}\,{A^T} + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = m\). Put the values in\(\dim \,{\rm{Col}}\,A = {\rm{rank}}\,{A^T}\) to get

\(\begin{array}{c}{\rm{rank}}\,{A^T} + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = m\\{\rm{dim}}\,{\rm{Col}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = m.\end{array}\)

Hence, the equalities in parts (a) and (b) are justified.