Let V be an infinite dimensional real vector space with a basis e1,e2,e3 . . . and let End(V) denote the set of linear transformations of T, i.e.,End(V) = {T : V → V | T is a linear transformation}.Define the elements A and B of End V byA(ei) = ei+1 for i = 1, 2, . . . ; B(e1) = 0, B(ei) = ei−1 for i = 2, 3, . . . .(a) Show that End(V ) is a ring with the usual addition and composition as multiplication.(b) Show that BA = 1End(V). Are A and B inverse to each other?(c) Assume that AC = 0End(V). Prove that C = 0End(V). Give an example of D ≠ 0End(V) such that DA = 0End(V).

Given that be an infinite dimensional real vector space with basis .Let denote the set of linear…

Answered: Let V be an infinite dimensional real…

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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Question

Let V be an infinite dimensional real vector space with a basis e₁,e₂,e₃ . . . and let End(V) denote the set of linear transformations of T, i.e.,
End(V) = {T : V → V | T is a linear transformation}.
Define the elements A and B of End V by
A(e_i) = e_i+1 for i = 1, 2, . . . ; B(e₁) = 0, B(e_i) = e_i−1 for i = 2, 3, . . . .
(a) Show that End(V ) is a ring with the usual addition and composition as multiplication.
(b) Show that BA = 1_End(V). Are A and B inverse to each other?
(c) Assume that AC = 0_End(V). Prove that C = 0_End(V). Give an example of D ≠ 0_End(V) such that DA = 0_End(V).