Q30E Consider an ellipse E聽in R2cent... [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 5: Q30E (page 262)

Consider an ellipse Ein $R^{2}$ centered at the origin. Show that there is an inner product $< ., . >$ in $R^{2}$ such that E consists of all vectors $\overset{鈫�}{x}$ with $| | \overset{鈫�}{x} | | = 1$ , where the norm is taken with respect to the inner product $< ., . >$ .

Short Answer

Expert verified

It is proved that there is an inner product $<., .>$ in $R^{2}$ such that E consists of all vectors $\overset{鈫�}{x}$ with $| | \overset{鈫�}{x} | | = 1$ , where the norm is taken with respect to the inner product $<., .>$ .

Step by step solution

Step by step solution Step 1: An inner product R2

Let $E = \{[\begin{matrix} x \\ y \end{matrix}] 鈭� R^{2} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\}$

We will define a liner mapping which send ellipse to a circle. Define the linear mapping
$T : R^{2} 鈫� R^{2}, [\begin{matrix} x \\ y \end{matrix}] 鈫� [\begin{matrix} x / a \\ y / b \end{matrix}]$

Observe that, T is a linear mapping and T maps ellipse to the unit circle.

Ker $T = \{\overset{鈫�}{0}\}$

Proof

Now, an inner product on $R^{2}$ :

$<\overset{鈫�}{x}, \overset{鈫�}{y}> = T \overset{鈫�}{x} 路 T \overset{鈫�}{y}$

From the exercise 5.5.17:

$\begin{array}{rcl} ||\overset{鈫�}{x}|| & = & <\overset{鈫�}{x}, \overset{鈫�}{x}> \\ = & T (\overset{鈫�}{x}) 路 T (\overset{鈫�}{x}) \\ = & [\begin{matrix} \frac{x_{1}}{a} \\ \frac{x_{2}}{b} \end{matrix}] 路 [\begin{matrix} \frac{x_{1}}{a} \\ \frac{x_{2}}{b} \end{matrix}] \\ = & \frac{x_{1}^{2}}{a} + \frac{x_{2}^{2}}{b} \\ = & 1 \end{array}$

Hence, the norm is taken with respect to the inner product $<., .>$ is

role="math" localid="1659610758022" $T : R^{2} 鈫� R^{2}, [\begin{matrix} x \\ y \end{matrix}] 鈫� [\begin{matrix} \frac{x}{a} \\ \frac{x}{b} \end{matrix}]$

It is proved that there is an inner product $<., .>$ in $R^{2}$ such that E consists of all vectors $\overset{鈫�}{x}$ withrole="math" localid="1659610974581" $| | \overset{鈫�}{x} | | = 1$ , where the norm is taken with respect to the inner product $<., .>$ .

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

Consider an ellipse Ein $R^{2}$ centered at the origin. Show that there is an inner product $< ., . >$ in $R^{2}$ such that E consists of all vectors $\overset{鈫�}{x}$ with $| | \overset{鈫�}{x} | | = 1$ , where the norm is taken with respect to the inner product $< ., . >$ .

Short Answer

Step by step solution

Step by step solution Step 1: An inner product R2

Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Consider an ellipse Ein R2centered at the origin. Show that there is an inner product <.,.>in R2 such that E consists of all vectors x鈫�with ||x鈫�||=1, where the norm is taken with respect to the inner product <.,.>.

Short Answer

Step by step solution

Step by step solution Step 1: An inner product R2

Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

Consider an ellipse Ein $R^{2}$ centered at the origin. Show that there is an inner product $< ., . >$ in $R^{2}$ such that E consists of all vectors $\overset{鈫�}{x}$ with $| | \overset{鈫�}{x} | | = 1$ , where the norm is taken with respect to the inner product $< ., . >$ .