Q22E True or false, if the matrix of ... [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 4: Q22E (page 199)

True or false, if the matrix of a linear transformation T(with respect to some basis) is $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ , then there must exist a nonzero element f in the domain of T such that T(f)=3f.

Short Answer

Expert verified

False.

Step by step solution

Determine the basis.

Consider a linear transformation T such that matrix corresponding to some basis is $B = [\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ .

As the dimension of the matrix B is $2 脳 2$ implies the dimension of the domain is 2 .

Assume the domain V is spanned by $\{\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}\}$ .

Therefore, the basis is $\{\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}\}$ .

Determine the element.

The vector ${\overset{鈫�}{v}}_{1}$ is spanned as ${\overset{鈫�}{v}}_{1} = 1 . {\overset{鈫�}{v}}_{1} + 0 . {\overset{鈫�}{v}}_{2}$ means ${[{\overset{鈫�}{v}}_{1}]}_{B} = [\begin{matrix} 1 \\ 0 \end{matrix}]$ .

As ${[T (f)]}_{B} = B {[f]}_{B}$ , the image of at point ${\overset{鈫�}{v}}_{1}$ is defined as follows.

${[T ({\overset{鈫�}{v}}_{1})]}_{B} = B {[{\overset{鈫�}{v}}_{1}]}_{B}$

Substitute the values $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ for B and $[\begin{matrix} 1 \\ 0 \end{matrix}]$ for $[{\overset{鈫�}{v}}_{1}]$ in the equation ${[T ({\overset{鈫�}{v}}_{1})]}_{B} = B {({\overset{鈫�}{v}}_{1})}_{B}$ as follows.

${[T ({\overset{鈫�}{v}}_{1})]}_{B} = B {[{\overset{鈫�}{v}}_{1}]}_{B} {[T ({\overset{鈫�}{v}}_{1})]}_{B} = [\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] {[T ({\overset{鈫�}{v}}_{1})]}_{B} = [\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] {[T ({\overset{鈫�}{v}}_{1})]}_{B} = [\begin{matrix} 3 \\ 0 \end{matrix}]$

Further, simplify the equation as follows.

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

Therefore, there must exist a nonzero element f in the domain of T such that T(f)=3f .

Hence, the statement is true.

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 the matrix of a linear transformation T(with respect to some basis) is $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ , then there must exist a nonzero element f in the domain of T such that T(f)=3f.

Short Answer

Step by step solution

Determine the basis.

Determine the element.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Decision Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

True or false, if the matrix of a linear transformation T(with respect to some basis) is [3504], then there must exist a nonzero element f in the domain of T such that T(f)=3f.

Short Answer

Step by step solution

Determine the basis.

Determine the element.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Decision Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

True or false, if the matrix of a linear transformation T(with respect to some basis) is $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ , then there must exist a nonzero element f in the domain of T such that T(f)=3f.