Let (a,b) and (c,d) be two vectors in R^2. If ad?bc=0, show that they are linearly dependent. If ad?bc ? 0, show that they are linearly independent.
Question 2. Let f(x) = azt for a, b, c, d e C, ad - bc +0. [2]a) Show lim + f() = oo if c = 0. [2]b) With c#0, show lim + f() = ? and lim,--4f(x) = 0.
10. Mobius transformations. Let
a, b, c, d ad-bc 0 . The
function is called a Mobius transformation (or linear fractional
transformation). Show that
a) lim z->inf T(z) = inf if c=0;
b)kim z-> inf T(z) = a/c and lim z-> d/c T(z) = inf if
c0
*10. Möbius transformations. Let a,b,c,d EC with ad-bc70. The function T(2) = 2 a2 + b cz + d à (2 +-d/c) is called a Möbius transformation (or linear fractional transformation). Show that...
8. (10 points) Consider the general linear group GL(2,26) = {© a) | where a, b, c, d e Zes and ad – be + 0} (a) Determine the order of the group GL(2, Z5). (b) List a Sylow 5-subgroup of GL(2, Z5).
Q3 Consider the group (Z3 x Z3, +), where again Z3 x Z3 = {(a,b) a, b e Z3}, and we define addition by (a, b) + (c,d) = (a + cb+d). So, for instance, (1, 2) + (0,2)=(1,1). (a) List all of the subgroups of (Z3 X Z3, +). You should explain why your list is complete (i.e., why there are no subgroups other than the ones you have written). (b) Which of these subgroups are cyclic? You do...
Choose a generic formula for combination reaction: A) A+B=C B) AB=A+B C) A+BC=AC+B D) AB+CD=AD+CB OB OA
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the x-y plane for the case ad - bc > 0. (a) Sketch D in the z-y plane for the case ad-bc 〈 0. (c) Calculate the area of D. Show all working.
(7) Let 0くa 〈 b 〈 c 〈...
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
14. What is the image of the upper half-plane under a mapping of the form az + b a, b, c, d real; ad - bc < 0?
14. What is the image of the upper half-plane under a mapping of the form az + b a, b, c, d real; ad - bc
Point A Point B Point C Point D (3, 2, 1) (-2, 3, 1) (2, 1,-1) (0,-1,-2) Equation of the Distance from the point D to the plane ABC Distance from the point A to the straight line BC Distance between the straight lines BC and AD Distance from the point D to the plane ABC Distance from the point A to the straight line BC Distance between the straight lines BC and AD
Point A Point B Point C...