\documentclass[12pt]{article}
\usepackage{amsmath, amssymb, paralist}
\usepackage{fullpage}
%\usepackage{local-draftcopy}
\begin{document}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Hom}{{\textrm{Hom}}}
\newcommand{\diag}{{\textrm{diag}}}
\begin{center}
Math 101 Syllabus\\
Standard Text: Dummit and Foote: Abstract Algebra, Chapters 4, 5,
10, 11, 12
\end{center}
\begin{enumerate}
\item\ [4 days] Basic Linear Algebra:
\begin{compactenum}
\item (Assumed) Linear independence, span, basis, dimension, independent
sets extend to a basis, generating sets can be pared down to a
basis.
\item Coordinates and matrix of a linear transformation relative
to a basis, change of basis. Examples: projection onto a
hyperplane, rotations.
\item Row reduction, echelon form, and consequences: free
variables, pivot variables, kernel and image, rank-nullity
theorem for $T:\R^n \to \R^m$ (via free and pivot variables),
elementary row operations and invertibility. Parallel
comments for column operations. Given $A \in M_{m\times
n}(F)$, discuss representative of cosets $GL_m(F) A$, $A
GL_n(F)$, and $GL_m(F) A GL_n(F)$, the last as precursor to
Smith normal form.
\item Rank - Nullity (vector space form)
\item Foreshadow Smith normal form by considering $A \in
M_{m\times n}(\Z)$ and row and column operations (over $\Z$) to produce
the nice representative in $GL_m(\Z) A GL_n(\Z)$ (when $m=n$,
$\diag(d_1, \dots, d_n)$ with $d_i \in \Z$ and $d_i \mid
d_{i+1}$, $1\le i\le n-1$). Example: structure of $\Z^n/K$
where $K$ is a subgroup generated by a collection of vectors.
Interpret as linear map and use two sided equivalence to
produce a new basis so that $\Z^n/K \cong \Z/d_1\Z \oplus
\cdots \oplus \Z/d_n\Z$ ($d_i \mid d_{i+1}$) [foreshadowing
invariant factor theorem].
\end{compactenum}
\item\ [4 days] Modules: basic properties.
\begin{compactenum}
\item Definitions, examples (vector spaces, abelian groups, $T:
V\to V$ linear map to $k[x]$-module structure on $V$. Notion of
a $k$-algebra ($M_n(k)$, $k[x]$, $End_k(V)$, $k[G]$) and UMP:
given any $k$-algebra $A$ and $a\in A$ there is a unique
$k$-algebra map $k[x] \to A$ taking $x \mapsto a$.
\item Direct sums of modules (external and internal); spin off
internal direct product of groups. Discuss product and direct
sum of vector spaces, mapping properties. Define product and
coproduct of modules and their construction. Show
$\Hom_R(N, \prod M_\alpha) \cong \prod_\alpha \Hom_R(N,
M_\alpha)$,
$\Hom_R(\coprod_\alpha M_\alpha, N) \cong
\prod_\alpha\Hom_R(M_\alpha, N)$ and $End(k^n) =
\Hom(k^n,k^n)\cong M_n(End_k(k)) \cong M_n(k)$
\end{compactenum}
\item\ [3 days] Exact sequences of modules; split exact sequences via
sections or retractions (existence of section equivalent to
existence of a retraction). Free modules and their construction;
Short exact sequences with $0 \to N \to M \to F \to 0$ with $F$ free
split. Any $R$-module is the quotient of a free $R$-module (review
isomorphism theorems if needed). Localization of modules, connection
to exactness, action on direct sums; application: rank of a module
over an integral domain is the dimension of the localization over
the field of fractions.
\item\ [6 days] PIDs; Finitely generated modules over PIDs, invariant
factor and elementary divisor theorems, applications to rational and
Jordan canonical forms. Diagonalizability.
\item\ [1 day] Dual Modules (duality and free modules)
\item\ [2 days] Sesquilinear forms. Unitary, Hermitian operators,
unitary diagonalization. Real symmetric matrices and spectral
theorem.
\item\ [8 days] Group actions, G-set structure
theorem, class equation, $p$-groups symmetric group, conjugacy
classes in $S_n$, Sylow theorems, semidirect products and split
extensions, classifying groups of small orders.
\end{enumerate}
Optional topics:
\begin{compactenum}
\item\ [2 days] (optional) Bilinear forms, isometry groups,
connections to dual spaces.
\end{compactenum}
\end{document}