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Problem 20. In this question, we'll consider linear transformations in R² and R³ that transform familiar geometric shapes. (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse x² + 11 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 + 1½-½y²+1 z² 4 2 16 = 1. Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T: Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.

Problem 20. In this question, we'll consider linear transformations in R² and R³ that transform familiar geometric shapes. (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse x² + 11 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 + 1½-½y²+1 z² 4 2 16 = 1. Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T: Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.

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. (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse x² + 11 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 + 1½-½y²+1 z² 4 2 16 = 1.
Transcribed Image Text:Problem 20. In this question, we'll consider linear transformations in R² and R³ that transform familiar geometric shapes. (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse x² + 11 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 + 1½-½y²+1 z² 4 2 16 = 1.
Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T: Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.
Transcribed Image Text:Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T: Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.
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