eading Question 4.1.1. Which of the following maps is a group homomorphism? (Select all that ply.) Ca) p: (R, +) → (R, +), x → |x| b) p: (R – {0},·) → (R – {0}, '), x → \x| (c) p: (R – {0},·) → (R – {0},·), x → d) p: (R – {0},·) → (R – {0},·), x → 2|x|

Determine which of the following are homomorphisms. Justify your answers. 1 0 (a) o : R→ GL2(R) defined by ø(a) a 1 a b (b) : GL2(R) → R" defined by ø = ab. d a (c) ¢: GL2(R) → R" defined by ø = ad – bc. c d

{ (80) : a, b,ce z}. and for all such A E L let f, g, h: L→ Z be given by b f(A) = a, g(A) = tr(A), h(A) = det(A) (a) Show that f is a surjective homomorphism. (b) Show that g is additive, but not multiplicative. (c) Show that h is multiplicative, but not additive. 7. Let L =

Let's simulate the process of constructing a matrix for a 3D transformation that rotates 13 degrees about the axis n = [0.3, 0.2, 0.5]. First step: is the vector n a unit vector? (on the answers, 0.3^2 means 0.3-squared, and so on) O Yes, since 0.3 +0.2 + 0.5 = 1 O No, since 0.3^2 +0.2^2 + 0.5^2 1 O No, since 0.3^2 +0.2^2 + 0.5^2 = 1

plz with details

Determine if each of the following maps is a ring homomorphism. (a) f : R → R defined by f(x) %3D -x 1 -2 = (6 1) a ( (b) g : M2(IR) → M2(R) defined 1