Question

If the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?

Short answer

The dimension of the row space of A is 3.

Step-by-step solution

Describe the given data

Given that \({\rm{dim}}\,{\rm{Nul}}\,A = 2\).

Use the rank theorem

Bythe rank theorem, you get

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,A &= n\\{\rm{rank}}\,A + 2 &= 5\\{\rm{rank}}\,A &= 5 - 2\\ &= 3.\end{aligned}\)

Draw a conclusion

Note that \({\rm{rank}}\,A = \dim {\rm{Col}}A = \dim {\rm{Row}}A\). Hence, \(\dim {\rm{Row}}\,{\rm{A}} = 3\).