Suppose V is a vector space and T: V → V is a linear operator. A subspace UCV is said to be T-ambivalent if T(U) CU. Fix an arbitrary vo € V, and define C(vo) = span {Tk (vo) : ke N}. Show that C(vo) is a T-ambivalent subspace of V, and that it's the smallest T-ambivalent subspace con- taining vo. Recall: To show that C(vo) is the smallest T-ambivalent subspace containing vo, show that if UCV is another other T-ambivalent subspace containing Vo, then C(vo) CU.
Suppose V is a vector space and T: V → V is a linear operator. A subspace UCV is said to be T-ambivalent if T(U) CU. Fix an arbitrary vo € V, and define C(vo) = span {Tk (vo) : ke N}. Show that C(vo) is a T-ambivalent subspace of V, and that it's the smallest T-ambivalent subspace con- taining vo. Recall: To show that C(vo) is the smallest T-ambivalent subspace containing vo, show that if UCV is another other T-ambivalent subspace containing Vo, then C(vo) CU.
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:Suppose V is a vector space and T: V → V is a linear operator. A subspace UCV is said
to be T-ambivalent if T(U) CU.
Fix an arbitrary vo € V, and define C(vo) = span {Tk (vo) : ke N}. Show that C(vo)
is a T-ambivalent subspace of V, and that it's the smallest T-ambivalent subspace con-
taining vo.
Recall: To show that C(vo) is the smallest T-ambivalent subspace containing vo, show
that if UCV is another other T-ambivalent subspace containing Vo, then C(vo) CU.
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