2. Decide whether each of the following sets is a subgroup of G = {1,-1, i,...
6. Decide whether each of the following sets of quantum numbers are allowwed in the hydeogen atom. Briefly explain why they are allowed, or why not a. 3, /2, m -3 allowed not allowed b. 4,/ 3, m0 allowed not allowed 2,/ 2, m+1 c. allowed not allowed 7. Give the maximum number of electrons in an atom that can have these quantum numbers. Show your reasoning. a. 4 b. 3, m, + 2 c. 3,/= 3 d. n 5,/-1...
(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the multi- plicative group CX of nonzero complex numbers, and that the set H = {E1} is a finite subgroup of {+1, £i}. (b) Compute the index of H in G. (c) Compute the set of left cosets G/H.
thx 11. A subgroup H of a group G is called normal if for all r E G, the left coset rG is equal to the right coset Gr. In each of the following cases, define whether H is a normal subgroup of G You do not need to show it is a subgroup. (a) G-S3, H e, (1,2)) (b) G = GL(2, R) (with operation matrix multiplication); H = (c) G-U(Z2s) (with operation multiplication modulo 24); H-1,11 11. A...
5. Determine whether the following sets and operations form a group (a) S fa +b/2l a, b EQ) (0 under multiplication (b) S = {2E C I Izl = 1} under multiplication (c) S = {A E Mn(R) | A' = A} under addition (where At denotes the transpose of a matrix A) (d) S A E M,(R)| A A under multiplication (e) S- (AE M,(R) det A under addition (f S-AE Mn (R) | det A-1 under multiplication
Linear Algebra Problem 2: Decide for the following sets of vectors whether they are linear independant, a generating set or even a basis of R3
For each of the following sets, indicate whether it is a vector space. If so, point out a basis of it; otherwise, point out which vector-space property is violated. 1.The set V of vectors [2x, x2] with x R2. Addition and scalar multiplication are defined in the same way as on vectors. 2.The set V of vectors [x, y, z] R3 satisfying x + y + z = 3 and x − y + 2z = 6. Addition and scalar...
For each of the following relations on the set of all real numbers, decide whether or not the relation is reflexive, symmetric, antisymmetric, and/or transitive. Give a brief explanation of why the given relation either has or does not have each of the properties. (x, y) elementof R if and only if: a. x + y = 0 b. x - y is a rational number (a rational number is a number that can be expressed in the form a/b...
2. The center of a group G is the set (a) Prove that Z(G) is a subgroup of G, and that it is normal in G (b) Compute the center of the following groups: GG, Di D, Qs, At, Sa, and Dax Qs 2. The center of a group G is the set (a) Prove that Z(G) is a subgroup of G, and that it is normal in G (b) Compute the center of the following groups: GG, Di D,...
Please show the steps clearly? 1. Determine whether R-(0 under multiplication and C-(0 under multiplication are isomorphic : G - G from G to itself is called an automorphism of G. Let 2. An isomorphism : GG be an automorphism and consider H ={g€ G|(g) = g}. Show that H is a subgroup of G
1. Decide whether each of the following is an inner product space. Justify your answers. (i) V = Mnxn(R) with (A, B) = tr(AB). (ii) V = M2x2(C) with (A, B) = tr (iii) V = P(R) with (f,g) = f(1)g(1). (iv) V = P(R) with :((1 ;-) B-4). (v) V is the collection of continuous functions from (0, 1) to C, and (5.9) = 'rg() dt. 4.s)-(sat).