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Problem 20. In this question, we'll consider linear transformations in R2 and R3 that transform familiar geometric shapes. 12 (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse 1½ x² + 1½³ y² = 1. By "maps the circle to the ellipse", we mean that every vector lying on the circle is sent to a vector lying on the ellipse. Is your answer unique? Justify rigorously! (b) Find a linear transformation that maps the unit sphere x² + y²+z² = 1 to the ellipsoid x²+ y²+ z2 16 = 1.

Problem 20. In this question, we'll consider linear transformations in R2 and R3 that transform familiar geometric shapes. 12 (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse 1½ x² + 1½³ y² = 1. By "maps the circle to the ellipse", we mean that every vector lying on the circle is sent to a vector lying on the ellipse. Is your answer unique? Justify rigorously! (b) Find a linear transformation that maps the unit sphere x² + y²+z² = 1 to the ellipsoid x²+ y²+ z2 16 = 1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 44E
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Problem 20. In this question, we'll consider linear transformations in R2 and R3 that transform familiar geometric shapes. 12 (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse 1½ x² + 1½³ y² = 1. By "maps the circle to the ellipse", we mean that every vector lying on the circle is sent to a vector lying on the ellipse. Is your answer unique? Justify rigorously! (b) Find a linear transformation that maps the unit sphere x² + y²+z² = 1 to the ellipsoid x²+ y²+ z2 16 = 1.
Transcribed Image Text:Problem 20. In this question, we'll consider linear transformations in R2 and R3 that transform familiar geometric shapes. 12 (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse 1½ x² + 1½³ y² = 1. By "maps the circle to the ellipse", we mean that every vector lying on the circle is sent to a vector lying on the ellipse. Is your answer unique? Justify rigorously! (b) Find a linear transformation that maps the unit sphere x² + y²+z² = 1 to the ellipsoid x²+ y²+ z2 16 = 1.
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