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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".

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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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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