Question

Suppose a \({\bf{4}} \times {\bf{7}}\) matrix A has four pivot columns. Is \({\bf{Col}}\,A = {\mathbb{R}^{\bf{4}}}\)? Is \({\bf{Nul}}\,A = {\mathbb{R}^{\bf{3}}}\)? Explain your answers.

Short answer

\({\rm{Col}}\,A = {\mathbb{R}^4}\), \({\rm{Nul}}\,A \ne {\mathbb{R}^3}\)

Step-by-step solution

Check for Col A

Since A has fourpivot columns, \({\rm{Col}}\,A = {\mathbb{R}^4}\).

Check for Nul A

The dim Nul A =3, but Nul A is a subspace of \({\mathbb{R}^7}\). Therefore, \({\rm{Nul}}\,A \ne {\mathbb{R}^3}\),