8. Let V and W be the vector spaces and T : V → W be linear. (a) Prove that T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. (b) Suppose that T is one-to-one and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent. (c) Suppose 3 = {v1, t2, ., v,} is a basis for V and T is one-to-one and onto. Prove that T(3) = {T(v1), T(v2), .. ,T(v,)} is a basis for W.

8. Let V and W be the vector spaces and T : V → W be linear. (a) Prove that T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. (b) Suppose that T is one-to-one and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent. (c) Suppose 3 = {v1, t2, ., v,} is a basis for V and T is one-to-one and onto. Prove that T(3) = {T(v1), T(v2), .. ,T(v,)} is a basis for W.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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