Q39E In Exercises 37聽 through 42 , f... [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: Q39E (page 160)

In Exercises 37 through 42 , find a basis $J$ of ${鈩�}^{n}$ such that the $J - matrix$ of the given linear transformation T is diagonal.
Orthogonal projection T onto the line in ${鈩�}^{3}$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ .

Short Answer

Expert verified

The matrix is, $B = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}]$

Step by step solution

Consider the vector.

The vector is,

$\overset{鈫�}{V_{1}} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

Consider the conditions to obtain the reflection about the line spanned as follows:

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

Compute the matrix using linear combination.

Consider the vector:

$\overset{鈫�}{V_{1}} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

The orthogonal vectors will be,

$\overset{鈫�}{V_{2}} = [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] {\overset{鈫�}{V}}_{3} = [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}]$

The matrix will be,

$鈭� B = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}]$

Final answer.

The matrix is, $B = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] [\begin{matrix} - 3 \\ 0 \\ 1 \end{matrix}]$

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

In Exercises 37 through 42 , find a basis $J$ of ${鈩�}^{n}$ such that the $J - matrix$ of the given linear transformation T is diagonal.
Orthogonal projection T onto the line in ${鈩�}^{3}$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ .

Short Answer

Step by step solution

Consider the vector.

Compute the matrix using linear combination.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Probability and Statistics

Mechanics Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 37 through 42 , find a basisJ of 鈩�n such that theJ-matrix of the given linear transformation T is diagonal.Orthogonal projection T onto the line in鈩�3 spanned by[123].

Short Answer

Step by step solution

Consider the vector.

Compute the matrix using linear combination.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Probability and Statistics

Mechanics Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises 37 through 42 , find a basis $J$ of ${鈩�}^{n}$ such that the $J - matrix$ of the given linear transformation T is diagonal.
Orthogonal projection T onto the line in ${鈩�}^{3}$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ .