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Chapter 2: Q2.8-19E (page 93)

Determine which sets in Exercises 15-20 are bases for \({\mathbb{R}^{\bf{2}}}\) and \({\mathbb{R}^{\bf{3}}}\). Justify each answer.

\(\left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{8}}}\\{\bf{1}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right]\)

Short Answer

Expert verified

The vectors are not the basis for \({\mathbb{R}^3}\).

Step by step solution

01

Form a matrix using the given column vectors

\(A = \left[ {\begin{array}{*{20}{c}}3&6\\{ - 8}&2\\1&{ - 5}\end{array}} \right]\)

02

Analyze the matrix formed using the vectors

The columns of A cannot span \({\mathbb{R}^3}\) because A cannot have a pivot in every row. It means the columns are not the basis of \({\mathbb{R}^3}\).

So, the vectors are not the basis for \({\mathbb{R}^3}\).

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Most popular questions from this chapter

Show that \({I_n}A = A\) when \(A\) is \(m \times n\) matrix. (Hint: Use the (column) definition of \({I_n}A\).)

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

1. \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\E&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\)

Use the inverse found in Exercise 3 to solve the system

\(\begin{aligned}{l}{\bf{8}}{{\bf{x}}_{\bf{1}}} + {\bf{5}}{{\bf{x}}_{\bf{2}}} = - {\bf{9}}\\ - {\bf{7}}{{\bf{x}}_{\bf{1}}} - {\bf{5}}{{\bf{x}}_{\bf{2}}} = {\bf{11}}\end{aligned}\)

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

37. Construct a random \({\bf{4}} \times {\bf{4}}\) matrix Aand test whether \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\). The best way to do this is to compute \(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\) and verify that this difference is the zero matrix. Do this for three random matrices. Then test \(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^{\bf{2}}}\) the same way for three pairs of random \({\bf{4}} \times {\bf{4}}\) matrices. Report your conclusions.

[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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