/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q2.8-15Q Determine which sets in Exercise... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q2.8-15Q (page 93)

Determine which sets in Exercise 15-20 are bases for \[{\mathbb{R}^{\bf{2}}}\] or \[{\mathbb{R}^{\bf{3}}}\]. Justify your each answer.

15. \[\left[ {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{2}}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{{\bf{10}}}\\{ - {\bf{3}}}\end{array}} \right]\]

Short Answer

Expert verified

The set \[\left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{10}\\{ - 3}\end{array}} \right]\] is a basis for \[{\mathbb{R}^2}\]

Step by step solution

01

Find the determinant

Let \[A = \left[ {\begin{array}{*{20}{c}}5&{10}\\{ - 2}&{ - 3}\end{array}} \right]\]. Then,

\[\begin{array}{c}\det A = 5\left( { - 3} \right) - 10\left( { - 2} \right)\\ = - 15 + 20\\\det A = 5 \ne 0\end{array}\]

This implies that A is invertible.

02

Use the inverse matrix theorem

Here, A is invertible. This implies that the columns of A form alinearly independent set, and the columns span \[{\mathbb{R}^2}\] by using statements (a), (e), and (h) in the inverse matrix theorem.

03

Conclusion

Hence, the columns of A \[\left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{10}\\{ - 3}\end{array}} \right]\] form a basis for \[{\mathbb{R}^2}\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{5}}\\{ - {\bf{3}}}&{\bf{1}}\end{aligned}} \right)\) and \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{5}}}\\{\bf{3}}&k\end{aligned}} \right)\). What value(s) of \(k\), if any will make \(AB = BA\)?

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

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)

Suppose \(CA = {I_n}\)(the \(n \times n\) identity matrix). Show that the equation \(Ax = 0\) has only the trivial solution. Explain why Acannot have more columns than rows.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.