Q35E Let聽I=(v鈫�1v鈫�2v鈫�3) be any ... [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 3: Q35E (page 160)

Let $I = ({\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2} {\overset{鈫�}{v}}_{3})$ be any basis of ${鈩�}^{3}$ consisting of perpendicular unit vectors, such that ${\overset{鈫�}{v}}_{3} = {\overset{鈫�}{v}}_{1} 脳 {\overset{鈫�}{v}}_{2}$ . In Exercises 31through 36 , find the $I - matrix$ B of the given linear transformation T from ${鈩�}^{3}$ to ${鈩�}^{3}$ . Interpret geometrically.
$T (\overset{鈫�}{x}) = \overset{鈫�}{x} - 2 ({\overset{鈫�}{v}}_{1} . \overset{鈫�}{x}) {\overset{鈫�}{v}}_{2}$

Short Answer

Expert verified

The matrix is, $B = [\begin{matrix} 1 & 0 & 0 \\ - 2 |{|{\overset{鈫�}{v}}_{1}|}^{2}| & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ .

Step by step solution

Consider the vectors.

The vectors are,

$T ({\overset{鈫�}{v}}_{1}) = {\overset{鈫�}{v}}_{1} - 2 ({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{1}) {\overset{鈫�}{v}}_{2} = {\overset{鈫�}{v}}_{1} - 2 {\overset{鈫�}{v}}_{2} T ({\overset{鈫�}{v}}_{2}) = {\overset{鈫�}{v}}_{2} - 2 ({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2}) {\overset{鈫�}{v}}_{2} = {\overset{鈫�}{v}}_{2} - {\overset{鈫�}{v}}_{2} T ({\overset{鈫�}{v}}_{3}) = {\overset{鈫�}{v}}_{3} - 2 ({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{3}) {\overset{鈫�}{v}}_{2} = {\overset{鈫�}{v}}_{3}$

Compute the matrix by constructing columns.

The formula is, $T ({\overset{鈫�}{v}}_{1}) = A {\overset{鈫�}{v}}_{1}, T ({\overset{鈫�}{v}}_{2}) = A {\overset{鈫�}{v}}_{2}$

Compute the matrix

$T ({\overset{鈫�}{v}}_{1}) = [\begin{matrix} 1 \\ - 2 \\ 0 \end{matrix}] T ({\overset{鈫�}{v}}_{2}) = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] T ({\overset{鈫�}{v}}_{3}) = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$

Substitute these values in the formula.

$鈭� B = [\begin{matrix} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Final answer.

The matrix is, $B = [\begin{matrix} 1 & 0 & 0 \\ - 2 |{|{\overset{鈫�}{v}}_{1}|}^{2}| & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

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

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

Short Answer

Step by step solution

Consider the vectors.

Compute the matrix by constructing columns.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Decision Maths

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.