Note the following results:Theorem 1: Let T : R" –→ R" be a linear transformation and let A be the standardmatrix 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 isinvertible. Let T1 : R3 → R³ and T : R³ → R³ are two linear transformations such that thecompositions 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 matrixrepresentation of T2, then what are the standard matrix representations of T10T2 andT20T1?

It is given that, T1 : ℝ3→ℝ3 and T2 : ℝ3→ℝ3 be any two linear transformations such that T1oT2 and…

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

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 25EQ

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