150A Algebra
Monica Vazirani November 3, 2010
Midterm
Name:
ID:
Section:
1. (10 points) In parts (a)(b), give a careful denition of the terms in boldface. It is set up in
several cases that you can just complete the sentence.
a. Let : G G be a homomorphi
Math 150A  Fall 09  Homework 9 Selected Solutions
(Based on solutions prepared by Je Ferreira )
5.4.1 Prove that a discrete group G consisting of rotations about the origin is cyclic and is generated by
where is the smallest angle of rotation in G.
Pro
Math 150A  Fall 09  Homework 7&8 Selected Solutions
(Based on solutions prepared by Je Ferreira )
2.10.1 Let G be the group of invertible real upper triangular 2 2 matrices. Determine whether or not the
following conditions describe normal subgroups H o
Math 167 homework 3 solutions
October 20, 2010
2.1.22: For which righthand sides (nd a condition on b1 , b2 , b3 ) are these
systems solvable?
(a)
1
4
2
x1
b1
2
8
4 x2 = b2
1 4 2
x3
b3
r2 2r1 and r3 + r1 give the conditions b2 2b1 = 0 and b3 + b1 = 0 fo
Math 150A  Homework 3 Selected Solutions
(Based on solutions prepared by Je Ferreira )
2.3.2 Prove that the products ab and ba are conjugate elements in a group G.
Proof. Wee need to show the existence of an element c G such that ab = c(ba)c1 . Taking c
Math 167 homework 2 solutions
October 13, 2010
1.5.44: Find a 3 by 3 permutation matrix with P 3 = I
Find a 4 by 4 permutation matrix with P with P 4 = I .
0010
001
1 0 0 0
P = 1 0 0
P =
0 1 0 0
010
0001
1.6.2
(but not P = I ).
(a) Find the inverses of t
Math 228A
Homework 2
Due Friday, 10/22/08, 4:00
1. Use the standard 3point discretization of the Laplacian on a regular mesh to nd a numerical
solution to the PDEs below. Perform a renement study using the exact solution to compute
the error that shows t
Math 167 homework 1 solutions
October 6, 2010
1.3.14a: Construct a 3 by 3 system that needs two row exchanges to reach
a triangular form and a solution. There are many examples, but this is one:
3y 2z = 12
z
= 3
4x +3y
=3
1.3.14b: Construct a 3 by 3 syste
Math 228A
Homework 1
Due Tuesday, 10/12/10
1. Let L be the linear operator Lu = uxx , ux (0) = ux (1) = 0.
(a) Find the eigenfunctions and corresponding eigenvalues of L.
(b) Show that the eigenfunctions are orthogonal in the L2 [0, 1] inner product:
1
uv

cos20
sin20
By trig, the translation vector is 2a sin0(sin0, cos0), which is orthog to (cos0, sin0).
Also, note, above, any q on line L would give us the same translation vector (nothing
special about (a,0) except it's easy to compute with).
n is a sc