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Let T₁ R³ R³ and T₂ R³ R³ are two linear transformations such that the : : compositions T₁0T2 and T₂0T₁ are defined. Are T₁0T2 and T₂0T₁ again linear transfor- mations? Explain. If Yes, let A₁ be the standard matrix representation of T₁ and A₂ be the standard matrix representation of T2, then what are the standard matrix representations of T₁0T2 and T20T1?

Let T₁ R³ R³ and T₂ R³ R³ are two linear transformations such that the : : compositions T₁0T2 and T₂0T₁ are defined. Are T₁0T2 and T₂0T₁ again linear transfor- mations? Explain. If Yes, let A₁ be the standard matrix representation of T₁ and A₂ be the standard matrix representation of T2, then what are the standard matrix representations of T₁0T2 and T20T1?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section: Chapter Questions
Problem 17RQ
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Note the following results: Theorem 1: Let T : R" –→ R" be a linear transformation and let A be the standard matrix representation of T. Then (a) T is onto if and only if the columns of A span R". (b) T is one-one if and only if the columns of A are linearly independent. Theorem 2: If a linear transformation T : R" → Rn is one-one and onto, then it is invertible.
Transcribed Image Text:Note the following results: Theorem 1: Let T : R" –→ R" be a linear transformation and let A be the standard matrix representation of T. Then (a) T is onto if and only if the columns of A span R". (b) T is one-one if and only if the columns of A are linearly independent. Theorem 2: If a linear transformation T : R" → Rn is one-one and onto, then it is invertible.
Let T1 : R3 → R³ and T : R³ → R³ are two linear transformations such that the compositions T10T2 and T20T1 are defined. Are T10T2 and T20T1 again linear transfor- mations? Explain. If Yes, let A1 be the standard matrix representation of T1 and A2 be the standard matrix representation of T2, then what are the standard matrix representations of T10T2 and T20T1?
Transcribed Image Text:Let T1 : R3 → R³ and T : R³ → R³ are two linear transformations such that the compositions T10T2 and T20T1 are defined. Are T10T2 and T20T1 again linear transfor- mations? Explain. If Yes, let A1 be the standard matrix representation of T1 and A2 be the standard matrix representation of T2, then what are the standard matrix representations of T10T2 and T20T1?
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