Q53E Sketch the image of the unit cir... [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: Q53E (page 75)

Sketch the image of the unit circle under the line transformation
$T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$

Short Answer

Expert verified

The image of circle under $T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$ is an ellipse.

Step by step solution

Compute the functions

Consider the matrix.

$T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$

The vector that has points on the unit circle is,

$C = {(\begin{array}{l} \cos t \\ \sin t \end{array}) : 0 鈮� t 鈮� 2 蟺}$

Let the vector be,

$\overset{鈫�}{x} = (\begin{array}{l} \cos t \\ \sin t \end{array})$

The linear transformation is,

$\begin{array}{l} T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) (\begin{array}{l} \cos t \\ \sin t \end{array}) \\ T (\overset{鈫�}{x}) = (\begin{array}{l} 5 \cos t \\ 2 \sin t \end{array}) \\ T (\overset{鈫�}{x}) = \cos t (\begin{array}{l} 5 \\ 0 \end{array}) + \sin t (\begin{array}{l} 0 \\ 2 \end{array}) \end{array}$

Consider the transformation

The transformation of the circle is,

$T (C) = {\cos t (\begin{array}{l} 5 \\ 0 \end{array}) + \sin t (\begin{array}{l} 0 \\ 2 \end{array}) : 0 鈮� t 鈮� 2 蟺}$

The above equation represents the ellipse with the axes given as,

$\overset{鈫�}{e_{1}} = [\begin{array}{l} 5 \\ 0 \end{array}], \overset{鈫�}{e_{2}} = [\begin{array}{l} 0 \\ 2 \end{array}]$

As, $\overset{鈫�}{e_{1}} 脳 \overset{鈫�}{e_{2}} = [\begin{array}{l} 5 \\ 0 \end{array}] 脳 [\begin{array}{l} 0 \\ 2 \end{array}] = 0$ , thus, these are perpendicular to each other.

Sketch the graph

Consider the vectors.

$\overset{鈫�}{e_{1}} = [\begin{array}{l} 5 \\ 0 \end{array}], \overset{鈫�}{e_{2}} = [\begin{array}{l} 0 \\ 2 \end{array}]$

Consider the linear transformations.

$T (C) = {\cos t (\begin{array}{l} 5 \\ 0 \end{array}) + \sin t (\begin{array}{l} 0 \\ 2 \end{array}) : 0 鈮� t 鈮� 2 蟺}$

Consider the circle.

$C = {(\begin{array}{l} \cos t \\ \sin t \end{array}) : 0 鈮� t 鈮� 2 蟺}$

The graph of the same is,

Hence, the image of circle under $T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$ is an ellipse.

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91影视

Sketch the image of the unit circle under the line transformation
$T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$

Short Answer

Step by step solution

Compute the functions

Consider the transformation

Sketch the graph

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91影视

Sketch the image of the unit circle under the line transformationT(x鈫�)=(5002)x鈫�

Short Answer

Step by step solution

Compute the functions

Consider the transformation

Sketch the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Pure Maths

Geometry

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

Sketch the image of the unit circle under the line transformation
$T (\overset{鈫�}{x}) = (\begin{matrix} 5 & 0 \\ 0 & 2 \end{matrix}) \overset{鈫�}{x}$