Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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![画像:3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M a simple module? (b) State and prove Schur's Lemma for simple modules. (c) Let AM(K) and M = K" the natural A-module. (i) Show that M is a simple K-module. (ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a is a matrix in the centre of M, (K). [Recall that the centre, Z(M,(K)) == {a Mn(K) | ab M,,(K)}.] = ba for all bЄ (iii) Explain briefly why this means End1(M) K, assuming that Z(M,,(K))~ K as K-algebras. Is this consistent with Schur's lemma?](https://content.bartleby.com/qna-images/question/3011c556-643e-4a01-b0e0-55d8cf24eddf/ef6de67d-4eb6-4c2c-862b-4f2d97a7e8fe/ry995ii_thumbnail.png){kind=link}
Transcribed Image Text:3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End1(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
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