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Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If v₁, ..., Vk Є V are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁,..., Un is a basis of V if and only if L(v₁), ..., L(Un) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".

Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If v₁, ..., Vk Є V are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁,..., Un is a basis of V if and only if L(v₁), ..., L(Un) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".

Linear Algebra: A Modern Introduction
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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Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If v₁, ..., Vk Є V are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁,..., Un is a basis of V if and only if L(v₁), ..., L(Un) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".
Transcribed Image Text:Problem 22. Let V and W be vector spaces and let L: VW be a linear transformation. Prove the following: (a) If v₁, ..., Vk Є V are such that L(v₁), ..., L(Uk) are linearly independent, then V1, ..., Uk are linearly inde- pendent. (b) If L is injective and V1, ..., Uk Є V are linearly independent, then L(v₁), ..., L(Uk) are linearly independent. (c) If L is invertible, then v₁,..., Un is a basis of V if and only if L(v₁), ..., L(Un) is a basis of W. In other words, we can freely pass "basis information" between V and W. This is one of the many incarnations of the slogan "Invertible linear transformations perfectly preserve linear-algebraic information".
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