Algebra , Assignment 3, Solutions
Due, Friday, Oct 1 1. Let v be any nonzero vector in Rn and set H = cfw_A GLn (R) : Av = v for some real (That is, those A for which v is an eigenvector.) Prove that H is a subgroup of GLn (R). There are three parts: Ide
Algebra , Assignment 7
Solutions 1. Let G be a nite group and let a, b G with a b. [Here denotes conjugacy.] Show o(a) = o(b). Solution:So b = g1 ag so if an = e then bn = g1 an g = g1 g = e. Also a = gbg1 so if bn = e then an = gbn g1 = gg1 = e. As the p
Algebra , Assignment 1
Course website (bookmark!): www.cs.nyu.edu/cs/faculty/spencer/algebra/index.html 1. List with description the symmetries of the square. (There are eight of them.) Give the table for the products of the symmetries. Give the inverse o
Math 311 Tutorial Sept. 15, 2015
3r2
1. Is U = 1 s r, s, t R a subspace of R3 ? Justify your answer.
2t
Answer. No, U is not a subspace of R3 . ~03 U and U is closed under addition, but U is
not closed under scalar multiplication.
2. Either show that th
Math 311 Tutorial October 13, 2015
This is Example 9 from section 2.1 of the lecture notes (also exercise 2 from the exercise set
for chapter 2, section 1).
1. Let V = cfw_(x, y) : x, y R, with operations addition and scalar multiplication defined
as foll
Algebra , Assignment 2 Solutions
Due, Friday, Sept 24 1. In S3 (reminder, this is our standard notation for the permutations on cfw_1, 2, 3) show that there are four elements x satisfying x2 = e and three elements x satisfying x3 = e. 123 123 123 123 Solu
Algebra , Assignment 4, Solutions
1. In this problem, assume (important!) that G is an Abelian. Set H = cfw_g G : g5 = e. (Warning: Expressions such as x1/5 are not well dened. Do not use them!) (a) Show H is a subgroup of G. Point out where the assumptio
Algebra , Assignment 6 Solutions
1. Let : G G be an automorphism of G. Let x, y G with y = (x). (a) Assume xs = e. Prove y s = e. Solution:y s = (x)s = (xs ) = (e) = e (b) Assume xs = e. Prove y s = e. Solution:As above, y s = (xs ). An automorphism is in
Algebra , Assignment 8
Solutions 1. Three problems about manipulating products of cyclic groups. (a) Write Z2 Z2 Z2 Z9 Z5 Z25 as the product of cyclic groups Zai , 1 i s (you nd the s) with ai dividing ai+1 for all 1 i < s. Solution:We line up each power
Algebra , Assignment 9
Solutions 1. Let R be a ring. Call a R a unit if ab = 1 for some b R. Let X be the set of units Prove that X forms a group under multiplication. What is our standard notation for X in the case where R = Zn ? Solution: Identity: As 1
Algebra , Assignment 10
Solutions 1. Let R be a ring and M R an ideal. Assume M = R (but do not assume M is maximal). (a) Assume there exists an ideal N with M N R and N = M, R. Let a N with a M . Prove that a has no multiplicative inverse in R/M Solution
Algebra , Assignment 11 SOLUTIONS
1. Call R a Turkey-Yam Ring if it is an Integral Domain and so that it has a size function d : R cfw_0 cfw_0, 1, 2, . . . with d() d( ) and with the following property: For all , R cfw_0 with not dividing there exist m,
Algebra , Assignment 12
Solutions I wasnt sure anymore and I will tell you, it is a strange process to feel ones mind changing, allowing ideas into your brain which it had once considered unthinkable. I cannot say its painful, or particularly pleasurable,
Algebra , Assignment Solutions
Not to be Submitted 1. With K = Z2 [x]/(x3 + x +1) create orthogonal latin squares L1 = (aij ) with aij = i + j and L2 = (bij ) with bij = i + xj . Associate K with cfw_0, . . . , 7 by setting x = 2. Create a Magic Square of