Question

In Exercise 12, give integers p and q such that Nul A is a subspace of \[{\mathbb{R}^p}\] and Col A is a subspace of \[{\mathbb{R}^q}\].

12. \[A = \left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{3}}\\{\bf{4}}&{\bf{5}}&{\bf{7}}\\{ - {\bf{5}}}&{ - {\bf{1}}}&{\bf{0}}\\{\bf{2}}&{\bf{7}}&{{\bf{11}}}\end{array}} \right]\]

Short answer

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

Thus, Nul A is the subspace of \[{\mathbb{R}^3}\].

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 \[4 \times 3\] matrix.

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

Conclusion

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