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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned 9 by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = =Ax for all vectors & ER³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21 • The vectors 3 and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line. Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If 1,..., Uk EV are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁, ..., Un is a basis of V if and only if L(v₁), ..., L(vn) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".

Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned 9 by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = =Ax for all vectors & ER³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21 • The vectors 3 and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line. Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If 1,..., Uk EV are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁, ..., Un is a basis of V if and only if L(v₁), ..., L(vn) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned 9 by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = =Ax for all vectors & ER³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21 • The vectors 3 and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
Transcribed Image Text:Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned 9 by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = =Ax for all vectors & ER³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21 • The vectors 3 and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If 1,..., Uk EV are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁, ..., Un is a basis of V if and only if L(v₁), ..., L(vn) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".
Transcribed Image Text:Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If 1,..., Uk EV are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁, ..., Un is a basis of V if and only if L(v₁), ..., L(vn) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".
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