Question

Justify the following equalities:

a.\({\rm{dim Row }}A{\rm{ + dim Nul }}A = n{\rm{ }}\)

b.\({\rm{dim Col }}A{\rm{ + dim Nul }}{A^T} = m\)

Short answer

The equalities in parts (a) and (b) are justified.

Step-by-step solution

Assume an arbitrary system and relate the given statement with it

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, \(\dim \,{\rm{Col}}\,A = {\rm{rank}}\,{A^T}\).

Use the rank theorem

a. By the rank theorem, \({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\). Put the values in \(\dim \,{\rm{Row}}\,A = {\rm{rank}}\,A\) to get

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

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

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

Draw a conclusion

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