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 |
Symmetries 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 |
Review | ||
| Friday February 17 |
MIDTERM I | MIDTERM I | MIDTERM I |
| 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 Characters. |
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 |
Review |
||
| 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 |
Review |
MIDTERM II | MIDTERM II Due at the beginning of class |
| March 27--31 | SPRING BREAK |
SPRING BREAK |
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 |
Review | ||
| Wednesday May 3 |
Review | ||
| Friday May 5 |
Review | 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) |
FINAL EXAM |
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: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cloud.sagemath.com/
or https://www.sharelatex.com/
for a cloud version.