Question

Which of the subspaces \({\rm{Row }}A\) , \({\rm{Col }}A\), \({\rm{Nul }}A\), \({\rm{Row}}\,{A^T}\) , \({\rm{Col}}\,{A^T}\) , and \({\rm{Nul}}\,{A^T}\) are in \({\mathbb{R}^m}\) and which are in \({\mathbb{R}^n}\) ? How many distinct subspaces are in this list?.

Short answer

There are four distinct subspaces in the list.

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 \(m \times n\). So, The null space in \(A\) must be the subspace of \({\mathbb{R}^n}\); the rows in \(A\) must be the subspace of \({\mathbb{R}^n}\); thecolumns in \(A\) must be the subspace of \({\mathbb{R}^m}\).

As the matrix \({A^T}\) has the size \(n \times m\), the null space in \({A^T}\) is a subspace of \({\mathbb{R}^m}\); the rows in \({A^T}\) are the subspace of \({\mathbb{R}^m}\); the columns in \({A^T}\) are the subspace of \({\mathbb{R}^n}\).

Draw a conclusion

As \({\rm{Row}}\,{A^T} = {\rm{Col}}\,A\) and \({\rm{Row}}\,A = {\rm{Col}}\,{A^T}\) , there are four possible distinct subspaces— \({\rm{Row}}\,A\),\({\rm{Col}}\,A\), \({\rm{Nul}}\,A\), \({\rm{Nul}}\,{A^T}\).