Q39E Can you find a line L聽in Rnand ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q39E (page 217)

Can you find a line Lin $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ such that $\overset{鈫�}{x} 路 pro j_{L} \overset{鈫�}{x}$ is negative? Explain, arguing algebraically.

Short Answer

Expert verified

For any $\overset{鈫�}{x} 鈭� R^{n}$ and line L inrole="math" localid="1659602164103" $R^{n}$ , we have $\overset{鈫�}{x} 路 p r o j_{L} (\overset{鈫�}{x}) 猢� 0$

Step by step solution

Step by step solution Step 1: Given:

The given condition is $\overset{鈫�}{x} 路 p r o j_{L} (\overset{鈫�}{x})$ is negative.

Find a line L in Rn

The answer is in the negative sense. That is, it is impossible to find such a line and a vector in $R^{n}$ .observe that, given any line L and vector $\overset{鈫�}{x} 鈭� R^{n}$ we have:

$\overset{鈫�}{x} 路 {proj}_{L} (\overset{鈫�}{x}) 鈯� {proj}_{L} (\overset{鈫�}{x}) 鈬� (\overset{鈫�}{x} 路 {proj}_{L} (\overset{鈫�}{x})) 路 {proj}_{L} (\overset{鈫�}{x}) = 0 鈬� \overset{鈫�}{x} 路 {proj}_{L} (\overset{鈫�}{x}) - | | {proj}_{L} (\overset{鈫�}{x}) | |^{2} 鈬� \overset{鈫�}{x} 路 {proj}_{L} (\overset{鈫�}{x}) = | | {proj}_{L} (\overset{鈫�}{x}) | | 猢� 0$

Hence,

For any $\overset{鈫�}{x} 鈭� R^{n}$ and line L in $R^{n}$ , there is $\overset{鈫�}{x} 路 {proj}_{L} (\overset{鈫�}{x}) 猢� 0$ .

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

Can you find a line Lin $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ such that $\overset{鈫�}{x} 路 pro j_{L} \overset{鈫�}{x}$ is negative? Explain, arguing algebraically.

Short Answer

Step by step solution

Step by step solution Step 1: Given:

Find a line L in Rn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Pure Maths

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Can you find a line Lin Rnand a vectorx鈫� in Rnsuch thatx鈫�路projLx鈫�is negative? Explain, arguing algebraically.

Short Answer

Step by step solution

Step by step solution Step 1: Given:

Find a line L in Rn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Pure Maths

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Can you find a line Lin $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ such that $\overset{鈫�}{x} 路 pro j_{L} \overset{鈫�}{x}$ is negative? Explain, arguing algebraically.