Q25E Which of the (nonlinear) transfo... [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: Q25E (page 97)

Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.25 $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{1}^{3} \\ x_{2} \end{matrix}]$ .

Short Answer

Expert verified

The inverse of the given transformation is $|\begin{matrix} x_{1} = \sqrt[3]{x_{1}} \\ x_{2} = y_{2} \end{matrix}|$ .

Step by step solution

Check whether the transformation is invertible or not

The given transformation is $|\begin{matrix} y_{1} = x_{1}^{3} \\ y_{2} = x_{2} \end{matrix}|$

We can rewrite the transformation as $f : R^{2} 鈫� R^{2}$ .such that

$f (x_{1}, x_{2}) = (x_{1}^{3}, x_{2})$

Then the given transformation is invertible if and only if is invertible.

(a) Suppose $f (x_{1}, x_{2}) = f (z_{1}, z_{2})$ for some $(x_{1}, x_{2}) = (z_{1}, z_{2}) 鈭� R^{2}$ .Then $x_{1}^{3} = z_{1}^{3}$ and $x_{2} = z_{2}$ .So $x_{1}, z_{1}$ are zero or nonzero together.

Now $(x_{1} - z_{1}) (x_{1}^{2} + x_{1} z_{1} + z_{1}^{2}) = x_{1}^{3} - z_{1}^{3} = 0$ (1)

Now $x_{1}^{2} + x_{1} z_{1} + z_{1}^{2} = 0 鈬� x_{1} z_{1} = (x_{1} + z_{1})$ and $- 3 x_{1} z_{1} = {(x_{1} - z_{1})}^{2}$ . So $x_{1} z_{1} = 0$ If $x_{1}, z_{1}$ are nonzero, then $x_{1}^{2} + x_{1} z_{1} + z_{1}^{2} = 0$ and hence from (1) $x_{1} = z_{1}$ .If $x_{1}, z_{1}$ are zero, then $x_{1} = z_{1}$ .Thus $x_{1} = z_{1}$ and $x_{2} = z_{2}$ . So,fis one-one.

(b) Let $(z_{1}, z_{2}) 鈭� R^{2}$ Then there exists $x_{1} 鈭� R$ such that $x_{1}^{3} = z_{1}$ Therefore $f ? (x_{1}, z_{2}) = (z_{1}, z_{2})$ . Hencefis onto.

Find the inverse of the given transformation

So,fis invertible and therefore the given transformation is invertible. Now . $f^{- 1} (z_{1}, z_{2}) = (\sqrt[3]{z_{1}}, z_{2})$ Therefore, inverse of the given transformation is $|\begin{matrix} x_{1} = \sqrt[3]{x_{1}} \\ x_{2} = y_{2} \end{matrix}|$ .

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

Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.25 $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{1}^{3} \\ x_{2} \end{matrix}]$ .

Short Answer

Step by step solution

Check whether the transformation is invertible or not

Find the inverse of the given transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Which of the (nonlinear) transformations from R2toR2to in Exercises 25 through 27 are invertible? Find the inverse if it exists.25 [y1y2]=[x13x2].

Short Answer

Step by step solution

Check whether the transformation is invertible or not

Find the inverse of the given transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Which of the (nonlinear) transformations from $R^{2} to R^{2}$ to in Exercises 25 through 27 are invertible? Find the inverse if it exists.25 $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} x_{1}^{3} \\ x_{2} \end{matrix}]$ .