4 (a)Show that the special orthogonal group SO(n) is a matrix group. You do NOT have to show it is amatrix Lie group.(b)Show that the symplectic groupISp(2n, R) := {A € M(2n, R) | A" JA = J} with J:=-I 0is a matrix group. You do NOT have to show it is a matrix Lie group.5Show that U(n) := {U € M(n,C) |U*U = I} is a matrix Lie group. What is the real dimension (i.e.,dimension as a submanifold of R' for some l e N) of U(n)?

We will use basic knowledge of group theory and linear algebra to answer this question.

Answered: (a) Show that the special orthogonal…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 24E

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