Q19E Find the matrices of the 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 2: Q19E (page 72)

Find the matrices of the linear transformations from R³to R³given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
19. The orthogonal projection onto the x-y-plane.

Short Answer

Expert verified

The matrix that represents the orthogonal projection onto x-y-plane is, $A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})$

Step by step solution

Compute the vector.

Consider the vector

$\begin{array}{l} \overset{鈫�}{x} = x \overset{鈫�}{m_{1}} + y \overset{鈫�}{m_{2}} + z \overset{鈫�}{m_{3}} \\ 鈭� \overset{鈫�}{x} = x [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}] + y [\begin{array}{l} 0 \\ 1 \\ 0 \end{array}] + z [\begin{array}{l} 0 \\ 0 \\ 1 \end{array}] \end{array}$

Consider the system.

The system is,

$\begin{array}{l} T (\overset{鈫�}{x}) = T (x \overset{鈫�}{m_{1}} + y \overset{鈫�}{m_{2}} + z \overset{鈫�}{m_{3}}) \\ 鈬� T (\overset{鈫�}{x}) = x T (\overset{鈫�}{m_{1}}) + y T (\overset{鈫�}{m_{2}}) + z T (\overset{鈫�}{m_{3}}) \end{array}$

Compute the matrix.

As $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ ,

Thus,

$(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}) [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{l} x \\ y \\ 0 \end{array}]$

Here, $z = 0$ represents the x-y plane.

$\begin{array}{l} T (\overset{鈫�}{x}) = x = (1, 0, 0) \\ T (\overset{鈫�}{y}) = y = (0, 1, 0) \\ T (\overset{鈫�}{z}) = (0, 0, 0) \end{array}$

Which implies that, $鈬� A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})$

Hence, $A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})$ is the matrix that represents the orthogonal projection onto thex-y-plane.

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

Find the matrices of the linear transformations from R³to R³given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
19. The orthogonal projection onto the x-y-plane.

Short Answer

Step by step solution

Compute the vector.

Consider the system.

Compute the matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Find the matrices of the linear transformations from R3to R3given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.19. The orthogonal projection onto the x-y-plane.

Short Answer

Step by step solution

Compute the vector.

Consider the system.

Compute the matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Find the matrices of the linear transformations from R³to R³given in Exercises 19through 23. Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
19. The orthogonal projection onto the x-y-plane.