5. .n by matrices with real entries, and the group operation is matrix multiplica-tion. Further recall that it has a subgroup SL,(R), consisting of matrices withdeterminant 1.Recall from class that the group GLn(R) is the group of invertible(a) Show the determinant map det :homomorphism.GL.(R) → (R \ {0}, ·) is a surjective(b) Justify why the kernel of the det map is SL,(R).(c) Using an important theorem from class, show that the quotient groupGL,(R)/SL,(R) is isomorphic to (R \ {0}, ·).

Answered: Image /qna-images/answer/4fc6e823-943e-46f2-ac0a-4c74d04bab59.jpg

Recall from class that the group GLn(R) is the group of invertible n by matrices with real entries, and the group operation is matrix multiplica- tion. Further recall that it has a subgroup SL,(R), consisting of matrices with determinant 1. (a) Show the determinant map det : homomorphism. (R) → (R \ {0}, ·) is a surjective (b) Justify why the kernel of the det map is SL(R). (c) Using an important theorem from class, show that the quotient group GL,(R)/SL,(R) is isomorphic to (R \ {0}, ·).

Recall from class that the group GLn(R) is the group of invertible n by matrices with real entries, and the group operation is matrix multiplica- tion. Further recall that it has a subgroup SL,(R), consisting of matrices with determinant 1. (a) Show the determinant map det : homomorphism. (R) → (R \ {0}, ·) is a surjective (b) Justify why the kernel of the det map is SL(R). (c) Using an important theorem from class, show that the quotient group GL,(R)/SL,(R) is isomorphic to (R \ {0}, ·).

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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