1 (5 points) Let A E GL(2, R). Compute the centralizer of A. Is the centralizer abelian? Explain. (61) =
Let A = [ 1 1 0 1] ∈ GL(2, R). Compute the centralizer of A. Is
the centralizer abelian? Explain.
Co 3. (5 points) Let A= e centralizer abelian? Explain. the centralizer of A. Is the
What is GL(3,R) / SL(3,R) isomorphic to?
GL(2, Z3) = {[a ] a, b,c,d € Zz, ad – bc = 0} What is the centralizer for at least three different elements in your group.
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
2 (2+2+1 marks) Consider the function GL(2,R-R A det A a) Prove that f is a surjective homomorphism. b) Verify that N-AL()dAE Ois a nomal subgroup of GL(2.R) GL(2.Ra group? a group? If so, with what operation? c) Is
EXERCISES 1. Let B be the standard Borel subgroup of GL(n, F). Determine all subgroups of SL(n, F) which contain BO SL(n, F)
EXERCISES 1. Let B be the standard Borel subgroup of GL(n, F). Determine all subgroups of SL(n, F) which contain BO SL(n, F)
4) Let G be a group and let a є G. The centralizer of a in G is defined as the set (i) Show that Ca(a) is a subgroup of G (ii) Find the centralizers of the elements r and y in the Dihedral group D4
Find the gradients of the following functions without writing
down a single partial derivative.
(c) f GL(n) - R defined by f(X) det(X).
(c) f GL(n) - R defined by f(X) det(X).
(GL and SL Entries; SL Trial Balance) The city of Asher had the following transactions, among others, in 20X7: 1. The council estimated that revenues of $210,000 would be generated for the General Fund in 20X7. The sources and amounts of expected revenues are as follows: Property Tax $150,000 Parking meters 5,000 Business licenses 30,000 Amusement licenses 10,000 Charges for services 8,000 Other revenues 7,000 $210,000 2. Property taxes of $152,000 were levied by the council; $2,000 of these taxes...
4. Find a subgroup of GL(3,R) that is isomorphic to 53. Hint consider what happens when you permute the three axes in R3 with elements of S3.