/*! 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} Q1E For each matrix Ain exercises 1 ... [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: Q1E (page 119)

For each matrix $A$ in exercises 1 through 13, find vectors that span the kernel of $A$ . Use paper and pencil.
$A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Short Answer

Expert verified

The kernel of $A$ is $\ker (A) = s p a n ([\begin{matrix} 0 \\ 0 \end{matrix}])$ .

Step by step solution

01

The kernel of a matrix

The kernel of a matrix $A$ is the solution set of the linear system role="math" localid="1660895259839" $A \overset{鈫�}{x} = 0$ .

02

Step 2: Find kernel of the given matrix

Solve the linear system $A \overset{鈫�}{x} = 0$ by reduced row-echelon form of $A$ :

$[\begin{matrix} 1 & 2 & 0 \\ 3 & 4 & 0 \end{matrix}] R_{2} 鈥� 鈫� 鈭� \frac{1}{2} R_{2} + \frac{3}{2} R_{1} [\begin{matrix} 1 & 2 & 0 \\ 1 & 1 & 0 \end{matrix}] R_{1} 鈫� R_{1} 鈭� 2 R_{2} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}]$

From the above calculation, we can say that the solution set of the linear system is $[\begin{matrix} 0 \\ 0 \end{matrix}]$ .

Thus, $\ker (A) = s p a n ([\begin{matrix} 0 \\ 0 \end{matrix}])$ .

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

Can you find a $3 脳 3$ matrix $A$ such that $i m (A) = k e r (A)$ ? Explain.

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 $k$ is any nonzero constant, then the equations $f (x, y) = 0$ and $k f (x, y) = 0$ define the same cubic.

45. Show that the cubic through the points $(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (0, 2) 补苍诲鈥� (0, 3)$ can be described by equations of the form $c_{5} x y + c_{8} x^{2} y + c_{9} x y^{2} = 0$ , where at least one of the coefficients $c_{5}, c_{8}, 补苍诲鈥� c_{9}$ is nonzero. Alternatively, this equation can be written as $x y (c_{5} + c_{8} x + c_{9} y) = 0$ . Describe these cubic geometrically.

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

46. $(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (0, 2), (0, 3), (1, 1)$ .

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Find a basis of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 1 \end{matrix}] for \overset{鈫�}{x} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$ .

Explain why you need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. See Theorem 3.3.4b.

See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision 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.