/*! 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} Q12E In Exercises聽聽through聽 , use... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q12E (page 131)

In Exercises through , use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.
12. $[\begin{matrix} 2 \\ 1 \end{matrix}], [\begin{matrix} 6 \\ 3 \end{matrix}]$ .

Short Answer

Expert verified

The vectors $[\begin{matrix} 2 \\ 1 \end{matrix}], [\begin{matrix} 6 \\ 3 \end{matrix}]$ are redundant and linearly dependent.

Step by step solution

01

Consider the set.

The vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . ., {\overset{鈫�}{v}}_{m}$ in ${鈩�}^{n}$ are linearly dependent if and only if there are nontrivial relation among them.

If linearly dependent, then, the vectors are said to be redundant.

Consider the vectors, $[\begin{matrix} 2 \\ 1 \end{matrix}], [\begin{matrix} 6 \\ 3 \end{matrix}]$

02

Check for linear independency of vectors.

The reduced form of the matrix is,

$(\begin{matrix} 2 & 6 \\ 1 & 3 \end{matrix}) = (\begin{matrix} 2 & 6 \\ 0 & 0 \end{matrix}) = (\begin{matrix} 1 & 3 \\ 0 & 0 \end{matrix})$

Solve the above equations.

$V_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}], V_{2} = [\begin{matrix} 3 \\ 0 \end{matrix}]$

$V_{2} = 3 v_{1}$

03

Final answer.

The vectors $[\begin{matrix} 2 \\ 1 \end{matrix}], [\begin{matrix} 6 \\ 3 \end{matrix}]$ are redundant and linearly dependent.

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

In Exercises 25through 30, find the matrix Bof the linear transformation $T (\overset{鈫�}{x}) = A (\overset{鈫�}{x})$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .

$A = (\begin{matrix} 0 & 1 \\ 2 & 3 \end{matrix}); {\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 1 \end{matrix}]$

Explain why fitting a cubic through the mpoints $P_{1} (x_{1}, y_{1}), . . . . . ., P_{m} (x_{m}, y_{m})$ amounts to finding the kernel of an mx10matrix A. Give the entries of theof row A.

Describe the images and kernels of the transformations in Exercises 23through 25 geometrically.

24. Orthogonal projection onto the plane $x + 2 y + 3 z = 0$ in ${鈩�}^{3}$ .

In Exercises 37 through 42 , find a basis $J$ of ${鈩�}^{n}$ such that the $J - matrix$ of the given linear transformation T is diagonal.

Orthogonal projection T onto the line in ${鈩�}^{3}$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ .

In Exercise 44 through 61, consider the problem of fitting a conic through $m$ given points $P_{1} (x_{1}, y_{1}), . ......, P_{m} (x_{m}, y_{m})$ in the plane. A conic is a curve in ${鈩�}^{2}$ that can be described by an equation of the form , $f (x, y) = c_{1} + c_{2} x + c_{3} y + c_{4} x^{2} + c_{5} x y + c_{6} y^{2} + c_{7} x^{3} + c_{8} x^{2} y + c_{9} x y^{2} + c_{10} y^{3} = 0$ where at least one of the coefficients $c_{i}$ is non zero. If is any nonzero constant, then the equations $f (x, y) = 0$ and $k f (x, y) = 0$ define the same cubic.

44. Show that the cubic through the points $(0, 0), (1, 0), (2, 0), (0, 1), and (0, 2)$ can be described by equations of the form $c_{5} x y + c_{7} (x^{3} 鈭� 3 x^{2} + 2 x) + c_{8} x^{2} y + c_{9} x y^{2} + c_{10} (y^{3} 鈭� 3 y^{2} + 2 y) = 0$ , where at least one of the coefficients is nonzero. Alternatively, this equation can be written as . $c_{7} x (x 鈭� 1) (x 鈭� 2) + c_{10} y (y 鈭� 1) (y 鈭� 2) + x y (c_{5} + c_{8} x + c_{9} y) = 0$

See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.