# 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.