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True or False? No justification is required. (a) If U is a subspace of a vector space V, then U' must be a subspace of V'. (b) If V is a finite-dimensional vector space, p E V', and U is a subspace of V containing null Q, then U must be equal to null p or V. (c) If V is a vector space, T E L(V), and v is an eigenvector of T, then v must be an eigenvector of T2. (d) If V is a finite-dimensional vector space, T E L(V), and v is a nonzero vector of V such that (T – 31)(T – 5I)(T – 71)v = 0, then v must be an eigenvector of T with eigenvalue 3, 5, or 7. (e) If V is a finite-dimensional vector space, T E L(V) is a diagonalizable linear operator, and å is the only eigenvalue of T, then T must be AI.

True or False? No justification is required. (a) If U is a subspace of a vector space V, then U' must be a subspace of V'. (b) If V is a finite-dimensional vector space, p E V', and U is a subspace of V containing null Q, then U must be equal to null p or V. (c) If V is a vector space, T E L(V), and v is an eigenvector of T, then v must be an eigenvector of T2. (d) If V is a finite-dimensional vector space, T E L(V), and v is a nonzero vector of V such that (T – 31)(T – 5I)(T – 71)v = 0, then v must be an eigenvector of T with eigenvalue 3, 5, or 7. (e) If V is a finite-dimensional vector space, T E L(V) is a diagonalizable linear operator, and å is the only eigenvalue of T, then T must be AI.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 74E: Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors...
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True or False? No justification is required. (a) If U is a subspace of a vector space V, then U' must be a subspace of V'. (b) If V is a finite-dimensional vector space, p E V', and U is a subspace of V containing null Q, then U must be equal to null p or V. (c) If V is a vector space, T E L(V), and v is an eigenvector of T, then v must be an eigenvector of T2. (d) If V is a finite-dimensional vector space, T E L(V), and v is a nonzero vector of V such that (T – 31)(T – 5I)(T – 71)v = 0, then v must be an eigenvector of T with eigenvalue 3, 5, or 7. (e) If V is a finite-dimensional vector space, T E L(V) is a diagonalizable linear operator, and å is the only eigenvalue of T, then T must be AI.
Transcribed Image Text:True or False? No justification is required. (a) If U is a subspace of a vector space V, then U' must be a subspace of V'. (b) If V is a finite-dimensional vector space, p E V', and U is a subspace of V containing null Q, then U must be equal to null p or V. (c) If V is a vector space, T E L(V), and v is an eigenvector of T, then v must be an eigenvector of T2. (d) If V is a finite-dimensional vector space, T E L(V), and v is a nonzero vector of V such that (T – 31)(T – 5I)(T – 71)v = 0, then v must be an eigenvector of T with eigenvalue 3, 5, or 7. (e) If V is a finite-dimensional vector space, T E L(V) is a diagonalizable linear operator, and å is the only eigenvalue of T, then T must be AI.
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