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1. Every linear transformation between finite dimensional vector spaces can be represented by a matrix. True or False? 2. The representing matrix of a linear transformation depends on the choice of bases for the domain and codomain. True or False? 3. There exists a linear transformation between finite dimensional vector spaces which has infinitely many different representing matrices. True or False? 4. There exists a linear transformation between finite dimensional vector spaces that has the same representing matrix no matter which bases are chosen for the domain and codomain. True or False? 5. Given two vector spaces, chosen bases in each, and a linear transformation between them, it then holds that the representing matrix of the composition of the linear transformation with itself is equal to the product of the representing matrix of the linear transformation with itself. True or False?

1. Every linear transformation between finite dimensional vector spaces can be represented by a matrix. True or False? 2. The representing matrix of a linear transformation depends on the choice of bases for the domain and codomain. True or False? 3. There exists a linear transformation between finite dimensional vector spaces which has infinitely many different representing matrices. True or False? 4. There exists a linear transformation between finite dimensional vector spaces that has the same representing matrix no matter which bases are chosen for the domain and codomain. True or False? 5. Given two vector spaces, chosen bases in each, and a linear transformation between them, it then holds that the representing matrix of the composition of the linear transformation with itself is equal to the product of the representing matrix of the linear transformation with itself. True or False?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Every linear transformation between finite dimensional vector spaces can be represented by a matrix. True or False? 2. The representing matrix of a linear transformation depends on the choice of bases for the domain and codomain. True or False? 3. There exists a linear transformation between finite dimensional vector spaces which has infinitely many different representing matrices. True or False? 4. There exists a linear transformation between finite dimensional vector spaces that has the same representing matrix no matter which bases are chosen for the domain and codomain. True or False? 5. Given two vector spaces, chosen bases in each, and a linear transformation between them, it then holds that the representing matrix of the composition of the linear transformation with itself is equal to the product of the representing matrix of the linear transformation with itself. True or False?
Transcribed Image Text:1. Every linear transformation between finite dimensional vector spaces can be represented by a matrix. True or False? 2. The representing matrix of a linear transformation depends on the choice of bases for the domain and codomain. True or False? 3. There exists a linear transformation between finite dimensional vector spaces which has infinitely many different representing matrices. True or False? 4. There exists a linear transformation between finite dimensional vector spaces that has the same representing matrix no matter which bases are chosen for the domain and codomain. True or False? 5. Given two vector spaces, chosen bases in each, and a linear transformation between them, it then holds that the representing matrix of the composition of the linear transformation with itself is equal to the product of the representing matrix of the linear transformation with itself. True or False?
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