Question

In Exercise 11, give integers p and q such that Nul A is a subspace of \[{\mathbb{R}^p}\] and Col A is a subspace of \[{\mathbb{R}^q}\].

11. \[A = \left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}&{ - {\bf{5}}}\\{ - {\bf{9}}}&{ - {\bf{4}}}&{\bf{1}}&{\bf{7}}\\{\bf{9}}&{\bf{2}}&{ - {\bf{5}}}&{\bf{1}}\end{array}} \right]\]

Short answer

Thus, the integers \[p = 4\] and \[q = 3\] such that Nul A is the subspace of \[{\mathbb{R}^p}\] and Col Ais a subspace of \[{\mathbb{R}^q}\].

Step-by-step solution

Use the definition of Nul A

By definition,Nul A is the set of all solutions of \[Ax = 0\]. When A has p columns, the solutions of \[Ax = 0\] belong to \[{\mathbb{R}^p}\]. Thus, Nul A is a subspace of \[{\mathbb{R}^p}\]. Note that the given matrixA has four columns.

Thus, Nul A is the subspace of \[{\mathbb{R}^4}\].

Use the definition of Col A

By definition, Col Ais the set of all linear combinations of its columns. This implies that the column space of an \[m \times n\] matrix is a subspace of \[{\mathbb{R}^m}\]. Note that the given matrix A is a \[3 \times 4\] matrix.

Thus, Col A is a subspace of \[{\mathbb{R}^3}\]

Conclusion

Thus, the integers \[p = 4\] and \[q = 3\] such that Nul A is the subspace of \[{\mathbb{R}^p}\] and Col A is a subspace of \[{\mathbb{R}^q}\].