Vectors, Matrices and Geometry, MATH 133.
Instructor: Dr. Eyal Z. Goren.
Time: MTWThF 9:0510:55
Location: BURN 1B23
Syllabus (section numbers refer
to Poole's book): Geometric vectors, dot product, lines and
planes
(Sections 1.1, 1.2, 1.3, Cross Products); Systems of linear equations,
spanning sets and linear independence (Sections 2.1, 2.2, 2.3);
Matrices,
matrix algebra, subspaces, basis, dimension, rank, linear
transformations
(Sections 3.1, 3.2, 3.3, 3.4, 3.5); Eigenvalues and eigenvectors,
determinants,
similarity and diagonalization (Sections 4.1, 4.2, 4.3, 4.4);
Orthogonality,
orthogonal complements and orthogonal projections, the GramSchmidt
process.
(Sections 5.1, 5.2, 5.3,5.4, 5.5(Quadratic Forms)).
Prerequisite: A course in
functions.
See Calendar for additional information.
Text Book: Linear Algebra,
by D. Poole.
Additional Texts: The text book
will suffice for everything in this course. You may however consult
also
the following texts if you need.
1) Anton, H. "Elementary linear algebra",
7th ed., or Anton & Rorres "Elementary linear algebra, applications
version", 8th ed., Chapters 17.
2) Nicholson, W. K., "Linear algebra with
applications", 2nd ed., Chapters 17.
Evaluation Method:
* WeBWorK
assignments 8%. There will be 4 Assignments posted every Thursday
with a deadline of Tuesday. Your WeBWorK assignments are on http://msr01.math.mcgill.ca/webwork/m133504/
* Written
assginments 7%. There will be 2 such Assignments to be given on
May
17 and May 25 and handed in by Friday May 22 and May 28, respectively.
Solutions
to Midterm. Midterm
Grades (out of 100)
* Midterm
exam (compulsory) 15%. To be given on Monday, May 17,
14:0016:00.
Room ARTS W120. Please verify this information and bring your
McGill
ID to the exam. The midterm will consist of one proof we did in class,
one easy proof not done in class and exercises as in the assignments.
* Final
Exam
70%. To be given on Friday, May 28, 9:0012:00, Room: LEA 14 & 15.
Please verify this information and bring your McGill ID to the exam.
The
structure of the final will be similar to the midterm's; there will be
some easy proofs, but most of the exam will consist of exercises very
similar
to those in the assignments and examples done in class. Final
Exam Copy
Office Hours: Tue, Th
11:0012:00.
(Burnside Hall, 1108; 5143983815).
Couse evaluations: Please fill
the course evaluations on line by loging in to your Minerva account.
The
deadline is Friday, May 28.
Help Desk: There is a "help
desk" which is a study center where advanced undergrads and
graduate
students are at hand to assist you. This is highly recommended.
Detailed Syllabus
(§ numbers refer to
Poole's book)
Date  Material  Comments  Written Assignments  Handouts 
May 37  Vectors:
geometric and algebraic;
adding and substracting vectors.
Dot product; lengths and angles. Lines and planes. Cross product. Linear equations. Methods for solving linear equations. Spanning sets and linear dependence. 
§§
1.11.3, 2.12.3
Students not familiar with complex numbers should read Appendix C in Poole. Knowledge of complex numbers is assumed from the fourth lecture. 
Solving
linear equations
New Version May 12 

May 1014  Algebra with matrices: addition, multiplication by scalar, multiplication, transpose. Elementary matrics. The inverse matrix and its calculation by rowreduction; application to linear equations. Subspace; row, column and null space of a matrix; basis and dimension. Linear transformations. Linear transformation and matrices. Rotations. Reflections. Composition.  §§ 3.13.5  Algebra with matrices  
May 1721  Projections. Inverse linear transformation. Eigenvalues and eigenvectors. Determinants. Laplace's expansions. Determinants of elementary matrices. The product (and other) formula for determinants. Calculation of determinants by row reduction. Determinant as a volume function. Cramer's rule and the adjoint matrix. The characteristic polynomial. Eigenvalues and eigenvectors revisited. Linear independence of eigenspaces. Similarity and diagonalization. Representing linear transformations in different bases and diagonalization. Applications.  §§ 4.14.4  Written Ass 1  
May 2428  Orthogonality in R^{n}. Orthonormal bases. Orthonormal matrices and distance preserving transformations. Orthogonal complements and orthogonal projections. The GramSchmidt process. A Symmetric matrice has real e.values and its e.spaces are orthogonal to each other. Orthogonal diagonalization. Applications: Quadratic forms and extrema of functions of 2 variables.  §§
5.15.5
May 24 is Victoria Day; makeup class is given ub 1B23 on May 25 and 26, 11:30  12:25. 
Written Ass 2  Diagonalization algoirthms 
Instructions for WebWork
Assignments:
These
weekly assignments will be available on the Web and will be answered on
the Web. Missed assignments cannot be redone. The WeBWorK assignments
are
worth 8% of your grade. Your WebWorK assignments are at http://msr01.math.mcgill.ca/webwork/m133504/
The initial login and password are your 9 digit student number. Please
change your password the first time you log in.
If you experience
technical
problems with WeBWorK please contact the administrator at
wwadmin@math.mcgill.ca.
Additional information about WeBWorK assignments can be found here.
Instructor: Dr. Eyal Z. Goren.
Time: MWF 8:359:25
Location: BURN 920.
Syllabus (in the large):
We shall cover part of Chapter 10 and
most of chapters 12, 13, 14 of Dummit and Foote.
January: Introduction to modules.
Modules over PID. Applications to linear transformations and finitely
generated
abelian groups. Time permitting: Smith's normal form.
FebruaryApril: Introduction to
filed theory. Algebraic and transcendental extensions; separable and
inseparable
extensions. Splitting fields and algebraic closure. Galois groups. The
fundamental theorem of Galois theory. Applications to solving equations
by radicals. Finite and cyclotomic fields. Time permitting:
Infinite
Galois groups. The inverse Galois problem.
Prerequisite:
MATH 251, MATH 370 (or equivalent courses
with my permission. Students that haven't taken a course on vector
spaces
can still enrol, given my permission, but are advised to catch up on
this
material by reading Dummit and Foote Sections 11.1  11.4).
Note: This course is normally taken
by honours students, though I do not consider that a requisite. One can
therefore expect it to be exciting, inspiring but also challenging.
Text Book:
*
Dummit
and Foote/ Abstract Algebra
(Third Edition), Wiley.
Other texts (on reserve at Schulich):
*
M. Artin / Algebra.
*
S. Lang / Algebra.
*
N. Jacobson / Abstract algebra
* I.
Stewart / Galois Theory
Evaluation Method:
* 20%
Assignments
(12 weekly, short assignments. Handedout and submitted on Mondays.
Submit
all. You may work together on your assignment, but in the end each has
to write his or her own solutions; identical assignments will be marked
as zero.)
*
25% Midterm (Tuesday,
March
9, 17:30  19:00 BURN 1120). The topics are General Theory of
Modules,
Modules over a PID and General Theory of Fields. The material includes
those parts of Chapters 10 and 12 covered in class, and sections 13.1 
13.4, 13.6 in Chapter 13 of Dummit and Foote. See detailed syllabus
below.
* 55% Final
Exam (Tuesday, April 20, 14:00, MAASS 328)
 If final exam grade is better
than midterm then midterm doesn't count. Assignment grades always count
(even in deferred/supplamental).
Office Hours: Wednesday,
Friday
9:3011:00. (Burnside Hall, 1108; 5143983815).
Detailed Syllabus
(§ numbers refer to Dummit and Foote)
Date  Material  Comments  Assignments and Solutions 
1/5  General
Theory of Modules:
Modules, submodules, morphisms of modules. Examples (in particular with base ring Z or F[x]). Kernels and quotient modules. The isomorphism theorems. 
§§ 10.110.2  
1/12  Chinese
Remainder Theorem.
Exact sequences. More examples. Free modules. Torsion and rank. Sum and
direct sum of modules.
Modules over a PID: The elementary divisors theorem for f.g. modules over a PID. 
§§ 10.210.3, § 12.1  Assignment 1 
1/19  The elementary divisors theorem (cont'd). The structure theorem for f.g. modules over a PID.  §§ 12.112.2  Assignment 2 
1/26  The structure
theorem for
f.g. modules over a PID (cont'd). The rational canonical form.
Application
to Jordan's canonical form.
General Theory of Fields: Fields  main examples, characteristic, F[x]/f(x), F(a); 
§§ 12.3, 13.1  Assignment 3 
2/2  Degree, algebraic elements, algebraic extensions.  §§ 13.1  13.2  Assignment 4 
2/9  Algebraic extension (cont'd). Straighthedge and compass constructions. The negative solution to doubling the cube, trisecting an angle, quadrature of the circle. Splitting fields and algebraic closure.  § 13.3  Assignment 5 
2/16  Splitting fields and algebraic closure (cont'd). Finite fields. Cyclotomic fields.  §§ 13.4  13.6  Assignment 6 
2/23  STUDY BREAK  
3/1  Cyclotomic
fields (cont'd).
Galois Theory. The automorphism group and subfields. 
Assignment 7  
3/8  Mideterm
on March 9, 17:3019:00, BURN 1120.
Definition of a Galois extension. A splitting field is Galois. Examples. Linear independence of characters. K/K^{G }is Galois with Galois group G. 
Assignment 8  
3/15  A Galois extension is a splitting field of a separable polynomial. The fundamental theorem of Galois theory.  Assignment 9  
3/22  Examples (finite fields and cyclotomic fields revisited). Construction of regular polygons. An S_{5} extension. Composite extensions.  Assignment 10  
3/29  The primitive element theorem. C is algebraically closed. Solvable and radical extensions. The insolvability of the quintic.  Assignment 11  
4/5  Galois groups of polynomials and the calculation of Galois groups over Q.  
4/13  Galois groups of polynomials and the calculation of Galois groups over Q (cont'd). 
Bonus problems.
You may solve and submit these problems
at any time before the last lecture. This will improve your assignment
grades. Partial solutions are allowed. Resubmittion is allowed.
1. Give an explicit example of a
real positive number r such that [Q(r):Q]
is a power of two but r is not constructible.
2. Let F = Z/pZ
a field with p elements and K an algebraic closure. Find all
subfields
of K.
3. Give an example of a polynomial
f(x) in Z[x] such that f(x) is irreducible
but
f(x)
is reducible modulo p for every prime p.
4. Let p>2 be a prime.
Prove
that the unique quadratic field K (i.e. [K:Q] =
2)
contained in Q(exp(2pi/p))
is Q(t^{1/2}) where t = (1)^{(p1)/2}p.
Time: MWF 8:359:25
Location: BURN 920.
Course Notes:
Here
are notes in pdf. The notes will be updated during the semester and
expanded to include all the material in group theory we'll cover in the
course. Be aware that notes may still contain typos; please let me know
if you find any!
Syllabus (in the large):
SeptemberOctober: Introduction
to groups, permutation groups; the isomorphism theorems for groups; the
theorems of Cayley, Lagrange and Sylow; structure of groups of low
order.
Time permitting: semidirect
products,
the simplicity of A_{n}, solvable groups.
NovemberDecember: Introduction
to ring theory; integral domains, fields, quotient field of an integral
domain; poynomial rings; unique factorization domains.
Time permitting: principal ideal
domains, Euclidean domains.
Prerequisite:
MATH 251 (or an equivalent course).
Note: This course is normally taken
by honours students, though I do not consider that a requisite
(approval
required). One can therefore expect it to be exciting, inspiring but
also
challenging.
Text Book:
Dummit and Foote/ Abstract Algebra
(Third
Edition), Wiley.
(Available at the bookstore; This is also
the textbook for MATH 371).
Other texts:
M. Artin/ Algebra.
J. Rotman/ Introduction to the theory
of groups.
Evaluation Method:
* 10%
Assignments
(11 weekly, short assignments. Handedout and submitted on Mondays.
Submit
all. You may work together on your assignment, but in the end each has
to write his or her own solutions; identical assignments will be marked
as zero.)
* 20%
Midterm
(to be conducted on Monday, November 3. Time and Place: 17:0518:25,
ARTS
210 )
* 70% Final
Exam. (to be conducted on Thursday, December 18, 9:00
 12:00, in ARTS W20)
 If final exam grade is better
than midterm then midterm doesn't count. Assignment grades always count
(even in deferred/supplamental).
Office Hours: Monday, Wednesday,
Friday 10:0011:00. (Burnside Hall, 1108; 5143983815).
Special Office hours prior to the exam:
I will be away on lectures and conferences during December 4  8 (until
noon) and 1319. However, during December 9  12, I will see students
each
day from 10:00  11:00.
Detailed Syllabus
Date  Material  Comments  Assignments and Solutions 
9/3  Introduction. Groups. Subgroups. Order of an element and the subgroup is generates. Subroup generated by a set. The groups Z, Z/nZ, Z/nZ*. The Dihedral group D_{2n}.  
9/8  The Symmetric group S_{n }(cycles, sign, transpositions, generators). The group GL_{n}(F). The quaternion group Q. Groups of small order. Direct products. The subgroups of (Z/2Z)^{2}. Cyclic groups and the structure of their subgroups. The group F* is cyclic. Commutator, centralizer and normalizer subgroups. Cosets.  Refresh your memory of the symmetric group.  Assignment
1
Solutions 
9/15  Cosets. Lagrange's Theorem. Normal subgroups and Quotient groups. Abelianization. Homomorphism, kernels and normal subgroups. The first homomorphism theorem.  In question 3) (2), p is a prime.  Assignment
2
Solutions 
9/22  The homomorphism theorems (cont'd). The lattice of subgroups. Group actions on sets: actions, stabilizers and orbits. Examples.  Assignment
3
Solutions 

9/29  Group actions on sets (cont'd): Cayley's theorem. The CauchyFrobenius formula. Applications to combinatorics: necklaces designs, 1415 square, Rubik's cube. Conjugacy classes in S_{n}.  Assignment
4
Solutions 

10/6  Conjugacy classes in A_{n}.The simplicity of A_{n}. The class equation. pgroups.  In question 1, the group G acts linearly on the vector space V.  Assignment
5
Solutions You can hand in your assignment 5 on Wednesday October 15. 
10/13  Free groups and Burnside's problem. Cauchy's Theorem. Syllow's Theorems  statement and examples.  Assignment
6
Solutions 

10/20  Syllow's Theorems  proof and applications (e.g., groups of order pq and p^{2}q). Finitely generated abelian groups.  This version > of the assignment correct typos of the one given in class.  Assignment
7
Solutions Number of Groups of order N 
10/27  Semidirect products and groups of order pq. Groups of order less than 16. Composition series. The Jordan Holder Theorem.  Assignment
8
Solutions 

11/3  Solvable groups. Rings  basics. Ideals and quotient rings. Examples: Z, Z/nZ, R[x], R[[x]], R((x)).  Midterm on Monday, November 3 17:0518:25, ARTS 210  Midterm
Solutions
Midterm Grades 
11/10  Examples: M_{n}(R), Quaternions. Creating new rings: quotient, adding a free variable, field of fractions. Ring homomorphisms. First isomorphism theorem. Behavior of ideals under homomorphisms.  In question 2, part (1), assume R is an integral domain!  Assignment
9
Solutions 
11/17  More on ideals: intersection, sum, product, generation, prime and maximal. The Chinese Remainder Theorem. Euclidean rings. Examples: Z, F[x], Z[i]. PID's. Euclidean implies PID. Greatest common divisor and the Euclidean algorithm.  Assignment
10
Solutions 

11/24  The Euclidean algorithm. Prime and irreducible elements + agree in PID. UFD's. Prime and irreducible agree in UFD. PID implies UFD. g.c.d. in a UFD. Gauss's Lemma.  Assignment
11
Solutions 

12/1  R UFD implies R[x] UFD. Existence of splitting fields. Construction of finite fields. 