Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q7E (page 131)

Consider a nonempty subset W of ${鈩�}^{n}$ that is closed under addition and under scalar multiplication. Is W necessarily a subspace of ${鈩�}^{n}$ ? Explain.

Short Answer

Expert verified

W is a subspace of ${鈩�}^{n}$

Step by step solution

Consider the set.

A subset W of the vector space ${鈩�}^{n}$ is called a (linear) subspace of ${鈩�}^{n}$ if it has the following three properties:

a. W contains the zero vector in ${鈩�}^{n}$ .

b. W is closed under addition: If ${\overset{鈫�}{w}}_{1}$ and ${\overset{鈫�}{w}}_{2}$ are both in W, then so is ${\overset{鈫�}{w}}_{1} + {\overset{鈫�}{w}}_{2}$ .

c. W is closed under scalar multiplication: If $\overset{鈫�}{w}$ is in W and k is an arbitrary scalar, then localid="1660801636209" $k \overset{鈫�}{w}$ is in W.

The nonempty subset W should satisfy the above conditions.

If W is a subset in which the localid="1660801628244" $0 鈭� W$ , as a result W must be a subspace of localid="1660801632742" ${鈩�}^{n}$ .

Final answer.

W is a subspace of ${鈩�}^{n}$ .

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Consider a nonempty subset W of ${鈩�}^{n}$ that is closed under addition and under scalar multiplication. Is W necessarily a subspace of ${鈩�}^{n}$ ? Explain.

Short Answer

Step by step solution

Consider the set.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Decision Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Consider a nonempty subset W of鈩�nthat is closed under addition and under scalar multiplication. Is W necessarily a subspace of鈩�n? Explain.

Short Answer

Step by step solution

Consider the set.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Decision Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Consider a nonempty subset W of ${鈩�}^{n}$ that is closed under addition and under scalar multiplication. Is W necessarily a subspace of ${鈩�}^{n}$ ? Explain.