Please solve 10.5. Recall that GL(n,R) denotes the general linear group and O(n) the orthogonalgroup. (See Theorem 6.18 and Exercise 6.13.) Define a map o: GL(n,R) → O(n) byP(A) = Q, where Q is the first factor in the QR-decomposition of A. (See the proof ofTheorem 8.4.) Prove the following assertions.(i) o is a retraction.(ii) O is homotopic to the identity mapping of M.

We will use the basic knowledge of algebraic topology to answer both the parts of this question…

10.5. Recall that GL(n, R) denotes the general linear group and O(n) the orthogonal group. (See Theorem 6.18 and Exercise 6.13.) Define a map o: GL(n,R) → O(n) by $(A) = Q, where Q is the first factor in the QR-decomposition of A. (See the proof of Theorem 8.4.) Prove the following assertions. %3D (i) o is a retraction. (ii) o is homotopic to the identity mapping of M.

10.5. Recall that GL(n, R) denotes the general linear group and O(n) the orthogonal group. (See Theorem 6.18 and Exercise 6.13.) Define a map o: GL(n,R) → O(n) by $(A) = Q, where Q is the first factor in the QR-decomposition of A. (See the proof of Theorem 8.4.) Prove the following assertions. %3D (i) o is a retraction. (ii) o is homotopic to the identity mapping of M.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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