Sebastian Casalaina

Homework and Syllabus

Abstract Algebra 2

MATH 4140/5140 Spring 2017

Homework is due in class and must be stapled, with your name and homework number on it, to receive credit.

Please read the suggested texts before class.

You may find the Mathematics Help Room (MATH 175) to be useful as a meeting point for discussing homework.

A * indicates that a homework assignment has not been finalized.

Date Topics Reading Homework
Wednesday January 18
Introduction to the course, and review of linear algebra
Vector spaces, linear maps, bases, dimension, direct sums, quotients.
M. Artin, Algebra (Second Edition), Prentice--Hall 2011.

Review Chapter 2

Review Chapter 3

The following .pdf gives a brief overview of vector spaces and linear maps.

Friday January 20
Review of linear algebra continued
Eigenvectors, the characteristic polynomial, triangular and diagonal forms, Jordan form.
Chapter 4

The following .pdf has a little more on computing Jordan forms.
HW 1

Artin Chapter 2 Exercises (pp.69--77):

4.3, 4.8, 5.1, 5.3, 5.6.

Artin Chapter 3 Exercises (pp.98--101):

4.1, 4.3, 6.1, 6.2.

On the .pdf, Exercises:

6.1.4, 6.1.7, 6.3.21, 6.3.22 
Monday January 23
Applications of linear operators
Orthogonal matrices and rotations.
Sections 5.1-2
Wednesday January 25
Applications of linear operators
The Cayley--Hamilton theorem, matrix exponential, applications.
Sections 5.3-4

For further reading on applications of exponentials and special forms of matrices, you may want to read:

Chapter 7 of T. Apostol, Calculus Volume II (Second Edition), Wiley 1969.

Sections 6.1-5 of M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 1974.

Friday January 27
Symmetry of plane figures, isometries, isometries of the plane.
Sections 6.1-3 HW 2

Artin Chapter 4 Exercises (pp.125--131):

1.3, 3.3, 3.4, 4.1, 4.2, 5.2, 5.6, 6.6, 7.1, 7.2, 7.6, 7.7.
Monday January 30
Symmetries continued
Finite groups of orthogonal operators on the plane, discrete groups of isometries.
Sections 6.4-5
Wednesday February 1
Symmetries continued
Groups operating on sets, counting formulas.
Sections 6.6-9
Friday February 3
Symmetries continued
Permutation representations, finite subgroups of the rotation group.
Sections 6.10-12 HW 3

Artin Chapter 5 Exercises (pp.150--153):

1.2, 1.5, 2.1, 2.3, 2.4, 3.2, 3.4, 4.3.
Monday February 6
Bilinear forms
Definition, symmetric forms, Hermitian forms.
Sections 8.1-3
Wednesday February 8
Bilinear forms continued
Orthogonality, Euclidean spaces, and Hermitian spaces.
Sections 8.4-5
Friday February 10
Bilinear forms continued
The spectral theorem.
Section 8.6 HW 4

Artin Chapter 6 Exercises (pp.188--194):

3.1, 3.2, 4.3, 5.3, 5.4, 7.1, 7.2, 7.8, 8.2, 10.2.
Monday February 13
Homework review

Wednesday February 15

Friday February 17
Monday February 20
Bilinear forms continued
Conics and quadrics, skew symmetric forms.
Sections 8.7-9
Wednesday February 22
Linear groups
The classical groups, spheres, the special unitary group.
Sections 9.1-3
Friday February 24
Linear groups continued
The rotation group, one-parameter groups.
Sections 9.4-5 HW 5

Artin Chapter 8 Exercises (pp.254--260):

1.1, 2.1, 3.2, 3.4, 4.3, 4.8, 4.9, 4.12, 5.1, 5.2, 5.4.
Monday February 27
Linear groups continued
The Lie algebra, translations in a group.
Sections 9.6-7
Wednesday March 1
Linear groups continued
Normal subgroups of the special linear group.
Section 9.8
Friday March 3
Group representations
Definitions, motivation.
Sections 10.1-2 HW 6

Artin Chapter 9 Exercises (pp.283--289):

3.2, 5.2, 5.4, 6.1, 6.7, 7.2, 8.5.

Due Monday March 6.
Monday March 6
Group representations continued
Definitions, characters, examples.

Wednesday March 8
Group representations continued
Morphisms of representations, subrepresentations.

Friday March 10
Group representations continued
Irreducible representations, direct sums, indecomposable representations.

HW 7

Artin Chapter 10 Exercises (pp.314--322):

1.1, 2.2, 3.2

Due Monday March 13.
Monday March 13
Group representations continued
Unitary representations
Section 10.3

Wednesday March 15
Group representations continued
Section 10.4
Friday March 17 Group representations continued
Characters continued.

HW 8

Artin Chapter 10 Exercises (pp.314--322):

2.3, 3.1, 3.4.
Monday March 20

Wednesday March 22
Review and hand out take home Midterm II

MIDTERM II will be handed out in class, and will be due at the beginning of class on Friday March 24.
Friday March 24

Due at the beginning of class

March 27--31 SPRING BREAK
Monday April 3
Review of exam

Wednesday April 5
Group representations continued
Characters continued.

Friday April 7
Group representations continued
One-dimensional characters.

Section 10.5 HW 9

Artin Chapter 10 Exercises (pp.314--322):

4.1, 4.3, 4.6
Monday April 10
Group representations continued
The regular representation.

Section 10.6

Wednesday April 12
Group representations continued
Schur's lemma.
Section 10.7
Friday April 14 Group representations continued
Proof of the orthogonality relations.
Section 10.8 HW 10

Artin Chapter 10 Exercises (pp.314--322):

4.8, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6.

Monday April 17
Group representations continued
Proof of the orthogonality relations continued.

Wednesday April 19
Group representations continued
Representations of the special unitary group.
Section 10.9
Friday April 21
Group representations continued
Representations of the special unitary group continued.

HW 11

Artin Chapter 10 Exercises (pp.314--322):

6.1, 6.9, 7.1.

Due Monday April 24
Monday April 24
Further topics

Wednesday April 26

Linear algebra continued
Tensor product and Hom.

Friday April 28
Linear algebra continued
Symmetric and alternating products.

HW 12

Review for the final exam
Monday May May 1

Wednesday May 3

Friday May 5
Take-home exam will be handed out in class. 

It is due at the beginning of the final exam scheduled on May 11.
Thursday May 11
FINAL EXAM 7:30 PM - 10:00 PM ECCR 118
(Lecture Room)


I strongly encourage everyone to use LaTeX for typing homework.  If you have a mac, one possible easy way to get started is with texshop. If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc. If you are using windows, you're on your own, but I'm sure there's something online. Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  This site can help you find LaTeX symbols by drawing:  You may also want to try or for a cloud version.