Q1E TRUE OR FALSE? If T聽is a linear... [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: Q1E (page 263)

TRUE OR FALSE? If Tis a linear transformation from $R^{n} to R^{n}$ tosuch that $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ are all unit vectors, thenmust be an orthogonal transformation.

Short Answer

Expert verified

The given statement is false, because the vectors must also be mutually orthogonal.

Step by step solution

Recollect the Orthogonal transformations and orthonormal bases theorem.

According to the orthogonal transformations and orthonormal bases theorem, a transformation Tfrom $R^{n}$ to $R^{n}$ is orthogonal if and only if the vectors role="math" localid="1661256767897" $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ form an orthonormal basis of $R^{n}$ .

Check whether the given statement is a true or false.

The given statement is if T is a linear transformation from $R^{n}$ to $R^{n}$ such that $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ are all unit vectors, thenmust be an orthogonal transformation.

It is given that the vectors $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ are all unit vectors.

But, by the above stated theorem, all the vectors must form an orthonormal basis for $R^{n}$ .

This means, the vectors must also be mutually orthogonal.

So, the given statement is false.

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

TRUE OR FALSE? If Tis a linear transformation from $R^{n} to R^{n}$ tosuch that $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ are all unit vectors, thenmust be an orthogonal transformation.

Short Answer

Step by step solution

Recollect the Orthogonal transformations and orthonormal bases theorem.

Check whether the given statement is a true or false.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

91影视

TRUE OR FALSE? If Tis a linear transformation fromRntoRntosuch thatT(e1鈫�),T(e2鈫�),.....,T(en鈫�)are all unit vectors, thenmust be an orthogonal transformation.

Short Answer

Step by step solution

Recollect the Orthogonal transformations and orthonormal bases theorem.

Check whether the given statement is a true or false.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

TRUE OR FALSE? If Tis a linear transformation from $R^{n} to R^{n}$ tosuch that $T (\overset{鈫�}{e_{1}}), T (\overset{鈫�}{e_{2}}), . . . . ., T (\overset{鈫�}{e_{n}})$ are all unit vectors, thenmust be an orthogonal transformation.