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Chapter 2: Q2.8-12Q (page 93)

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

Expert verified

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

01

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 \[{\mathbb{R}^3}\].

02

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 \[{\mathbb{R}^4}\].

03

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 \[{\mathbb{R}^q}\].

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Most popular questions from this chapter

Suppose P is invertible and \(A = PB{P^{ - 1}}\). Solve for Bin terms of A.

(M) Read the documentation for your matrix program, and write the commands that will produce the following matrices (without keying in each entry of the matrix).

  1. A \({\bf{5}} \times {\bf{6}}\) matrix of zeros
  2. A \({\bf{3}} \times {\bf{5}}\) matrix of ones
  3. The \({\bf{6}} \times {\bf{6}}\) identity matrix
  4. A \({\bf{5}} \times {\bf{5}}\) diagonal matrix, with diagonal entries 3, 5, 7, 2, 4

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. [Hint: Given u, v in \({\mathbb{R}^n}\), let \[{\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\]. Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \[T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \]. Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).]

Explain why the columns of an \(n \times n\) matrix Aare linearly independent when Ais invertible.

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]
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