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Suppose T and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not? Let A be the standard matrix for the linear transformation T andB be the standard matrix for the linear transformation U. Choose the correct answer below. O A. Yes, it is true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the trivial mapping and so AB = 0. This implies that either A or B is the zero matrix, and so BA = 0. This implies that U(T(x)) is also the trivial mapping. O B. No, it is not true. ABT is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so ABT =I. However, this does not imply that BAT = I, where BAT is the standard matrix for U(T(x)). So U(T(x)) is not necessarily the identity matrix. O C. Yes, it is true. AB is the standard matrix of the mapping x-T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB =I. Since both A and B are square and AB = I, the Invertible Matrix Theorem states that both A and B invertible, and B = A-1. Thus, BA = I. This means that the mapping x-U(T(x)) is the identity mapping. Therefore, U(T(x)) = x for all x in R". O D. No, it is not true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so AB = I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(T(x)), U(T(x)) is not necessarily the identity matrix.

Suppose T and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not? Let A be the standard matrix for the linear transformation T andB be the standard matrix for the linear transformation U. Choose the correct answer below. O A. Yes, it is true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the trivial mapping and so AB = 0. This implies that either A or B is the zero matrix, and so BA = 0. This implies that U(T(x)) is also the trivial mapping. O B. No, it is not true. ABT is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so ABT =I. However, this does not imply that BAT = I, where BAT is the standard matrix for U(T(x)). So U(T(x)) is not necessarily the identity matrix. O C. Yes, it is true. AB is the standard matrix of the mapping x-T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB =I. Since both A and B are square and AB = I, the Invertible Matrix Theorem states that both A and B invertible, and B = A-1. Thus, BA = I. This means that the mapping x-U(T(x)) is the identity mapping. Therefore, U(T(x)) = x for all x in R". O D. No, it is not true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so AB = I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(T(x)), U(T(x)) is not necessarily the identity matrix.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Suppose T and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not? Let A be the standard matrix for the linear transformation T andB be the standard matrix for the linear transformation U. Choose the correct answer below. O A. Yes, it is true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the trivial mapping and so AB = 0. This implies that either A or B is the zero matrix, and so BA = 0. This implies that U(T(x)) is also the trivial mapping. O B. No, it is not true. ABT is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so ABT =I. However, this does not imply that BAT = I, where BAT is the standard matrix for U(T(x)). So U(T(x)) is not necessarily the identity matrix. O C. Yes, it is true. AB is the standard matrix of the mapping x-T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB =I. Since both A and B are square and AB = I, the Invertible Matrix Theorem states that both A and B invertible, and B = A-1. Thus, BA = I. This means that the mapping x-U(T(x)) is the identity mapping. Therefore, U(T(x)) = x for all x in R". O D. No, it is not true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so AB = I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(T(x)), U(T(x)) is not necessarily the identity matrix.
Transcribed Image Text:Suppose T and U are linear transformations from R" to R" such that T(Ux) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not? Let A be the standard matrix for the linear transformation T andB be the standard matrix for the linear transformation U. Choose the correct answer below. O A. Yes, it is true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the trivial mapping and so AB = 0. This implies that either A or B is the zero matrix, and so BA = 0. This implies that U(T(x)) is also the trivial mapping. O B. No, it is not true. ABT is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so ABT =I. However, this does not imply that BAT = I, where BAT is the standard matrix for U(T(x)). So U(T(x)) is not necessarily the identity matrix. O C. Yes, it is true. AB is the standard matrix of the mapping x-T(U(x)) due to how matrix multiplication is defined. By hypothesis, this mapping is the identity mapping, so AB =I. Since both A and B are square and AB = I, the Invertible Matrix Theorem states that both A and B invertible, and B = A-1. Thus, BA = I. This means that the mapping x-U(T(x)) is the identity mapping. Therefore, U(T(x)) = x for all x in R". O D. No, it is not true. AB is the standard matrix for T(U(x)). By hypothesis, T(U(x)) = x is the identity mapping and so AB = I. However, matrix multiplication is not commutative, so BA is not necessarily equal to I. Since BA is the standard matrix for U(T(x)), U(T(x)) is not necessarily the identity matrix.
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