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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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