189251B Algebra II
Lecturer: Eyal Z. Goren Office:
1108
Tel:
3983815
Office Hours: Wednesday 13:0014:00 (in office), 17:0018:00
(Burn 1214).
Time of Course: MWF 8:309:30
Location of Lectures: Burnside Hall 1B24
Syllabus: Vector spaces, spanning sets, basis, dimesnion. Linear
maps and their matrix representations. Applications to systems of linear
equations. Determinants. Canonical forms. Duality. Bilinear and quadratic
forms. Inner product spaces. Diagonalization of selfadjoint operators.
Text book: Lipschutz, Seymour/Schaum's outline of theory
and problems of linear algebra, 3rd edition.
Additional references: Hoffman & Kunze/Linear Algebra,
(Prentice Hall).
; Lawson/Linear Algebra, (Wiley).
Dates of exams: Midterm Feb.
26, 6:007:30pm in ARTS 255. Final Thursday, April
26, 14:0017:00, BH
Method of Evaluation:  *Weekly assignments (15%): Best 8 out of 11. Handed out and collected on Fridays. 
*Midterm exam (20%, unless Final exam is better!).  
*Final exam (65%, unless better than midterm and in that case 85%).  
*Note carefully: Assignments grade is always 15% of the final grade, even if midterm or final exams are better. In case the right to write a supplemental exam is granted to a student, it would carry the same weight as the original final exam (i.e., 65%, unless better than midterm and in that case 85%), unless special circumstances arise. In that case, the student should approach me. 
Getting back your assignments:The
assignmnets will be left checked and marked in the course shelf in the
study center  Burnside Hall, 9th floor. Note that assignments sheets
are available from there also.
The assignments marking: In case
you believe that you should get a higher mark on your assignment, for whatever
reason, first resubmit your assignment with an explanatory note.
If you are not statisfied with the answer, then come to see me. Only in
very rare cases I will overrule the marker's decision.
Midterm: The material for the MidTerm
consists of all the material discussed in class and the corresponding sections
in the TextBook (even though the "core" material is what we did in class).
You also need to know the material of the assignments. Everything could
be on the Midterm. Don't make any assumptions!
The exam itself consists of three parts. Part
A consists of 5 multiple choice questions where you only need to give
the answer. Each question is worth 8 points. Part B consists of
2 questions. Each question is worth 30 points. You have to provide full
solutions. Part C is optional and is for extra credit only. It consists
of one question, which could be rather difficult without prior experience.
It will be worth 15 points at most.
Review. Tuesday,
April 17, 10:0012:00, Room 920 BH.
Special office hours. Tuesday
April 17, 13:0014:00. Tuesday April 24, 13:0014:00.
Structure of the final. Part
A: 7 questions of multiple choice (4 points
each). Part B: Answer 3 out of 4 questions (24 points each). Each
question has several parts. You would be required to know many proof and
be able to compute. A perfect knowledge of the class notes and the assignments
is a good preparation for this. Part C: optional (12 points). I
consider this harder than the rest of the exam.
GRADES
Detailed Syllabus
Week  Material  Assignment  Comments  Learn More! (No... it's not on the final!) 
Jan. 4  6  Overview of the course.
Vector spaces and subvector spaces: examples. 
None  *  History of abstract vector spaces. 
Jan. 812  Linear dependece. Spanning sets. Dimension.  Assignment 1: ps or pdf or dvi  Correction to Ass 1: ps or pdf or dvi  * 
Jan. 15 19  Coordinates and change of basis. Linear maps.  Assignment 2: ps or pdf or dvi  *  * 
Jan. 2226  Nilpotent operators and projections. Linear maps and matrices.  Assignment 3: ps or pdf or dvi  worksheet example 1  * 
Jan. 29 Feb.2  Linear maps and matrices. Detetminants. Permutations.  Assignment 4: ps or pdf or dvi  No office hours on January 31. Office hours for
this week: Monday 12, 56 (in room 1214).
A correction was introduced in the solution to question 4. Please correct the solutions in the handout I gave you. 
History of matrices and determinants. 
Feb. 59  The adjoint matrix. Systems of linear equations.  No Assignment.  worksheet example 2  Read about Fields medalists (The Nobel prize of Math). 
Feb. 1216  Systems of linear equations. Cramer's rule.
Friday: Review. 
Assignment 5: ps or pdf or dvi Submit Friday March 2. 
*  * 
Feb. 1923  Study Break  *  *  * 
Feb. 26March 2  Quotient spaces. The dual vector space. Dual basis. Duality theory.  Assignment 6: ps or pdf or dvi  worksheet example 3  MidTerm: ps
dvi
pdf
Solutions:ps dvi pdf 
March 59  Inner product spaces. CauchySchwartz inequality and applications. Orthonormal basese and GramSchmidt process.  Assignment 7: ps or pdf or dvi  worksheet example 4  * 
March 1216  Orthogonal projections and data fitting curves.
eigenvalues, eigenvectors and eigenspaces. The characteristic polynomial. Diagonalization and applications. 
Assignment 8: ps or pdf or dvi  *  * 
March 1923  CayleyHamilton theorem. The minimum polynomial. Criteria for diagonalizable.  Assignment 9: ps or pdf or dvi  worksheet example 5  Arthur Cayley 
March 2630  Invariant subspaces. Primary Decomposition Theorem. Jordan Canonical Form.  Assignment 10: ps or pdf or dvi  Note. Ass. 10 had several typos. The version here is corrected. You can submit by Monday April 9.  Camille Jordan 
April 26  Jordan Canonical Form. Diagonalization of self adjoint operators. Unitary operators.  Assignment 11: ps or pdf or dvi  *  * 
April 9  Rigid motions of the plane and space.  *  *  * 
189235A Algebra I
Lecturer: Eyal Z. Goren Office:
1108
Office
Hours: Mon 9:3010:30, Wed 1:00  2:00.
T.A.'s: M. Greenberg, M.H.
Nicole.
Time: MWF 8:309:30 (+1 hour tutorial)
Location: EngineeringMacDonald building, Room 280.
Syllabus: Sets and relations. Groups, Rings and fields. Integers,
rationals, real and complex numbers; modular arithmetic. Polynomials over
a field. Divisibility theory for integeres and polynomials. Linear equations
over a field. Introduction to vector spaces.
Text book: Thomas W. Hungerford: Abstract Algebra: An Introduction,
Saunders College Publishing. (Call Number: QA162 H86).
Midterm: Monday, October 16 , 7:30  8:45. Room:
LEA 26. There is no conflict with Cal III.
Method of Evaluation:  *Weekly assignments (15%): Best 8 out of 10. Handed out and collected on Fridays. 
*Midterm exam (20%, unless Final exam is better!).  
*Final exam (65%, unless better than midterm and in that case 85%).  
*Note carefully: Assignments grade is always 15% of the final grade, even if midterm or final exams are better. In case the right to write a supplemental exam is granted to a student, it would carry the same weight as the original final exam (i.e., 65%, unless better than midterm and in that case 85%), unless special circumstances arise. In that case, the student should approach me. 
Office hours: Monday 9:30  10:30, Wednesday 1:00  2:00.
Getting back your assignments:
The assignmnets will be left checked and marked in the course shelf in
the study center  Burnside Hall, 9th floor. Note that assignments sheets
are available from there also.
The assignment marking: In case
you believe that you should get a higher mark on your assignment, for whatever
reason, first contact your TA. If you are not statisfied with his answer,
then come to see the Instructor. Only in very rare cases I will overrule
the TA's decision.
Midterm: The material for the MidTerm
consists of all the material discussed in class (from Sep. 6 to Oct. 16)
and the corresponding sections in the TextBook (even though the "core"
material is what we did in class). You also need to know the material of
the assignments. Everything could be on the Midterm: from sets to arithmetic
of polynomial rings. Don't make any assumptions!
The exam itself consists of three parts. Part A consists of
5 multiple choice questions where you only need to give the answer. Each
question is worth 8 points. Generally speaking, these questions are at
the level of AB questions in your text book. Part B consists of
2 questions. Each question is worth 30 points. You have to provide full
solutions. The questions are at the level of difficulty BC of your text
book (i.e. like the harder questions in the assignments). Part C
is optional and is for extra credit only. It consists of one question,
which could be rather difficult without prior experience. I recommend attempting
it only if you have finished (or gave up) on the rest. It will be worth
15 points at most.
Office Hours:  Monday Dec. 4  09:3010:30 (Goren)  
Wednesday Dec. 6  13:0014:00 (Goren) 16:0018:00 (Nicole)  
Thursday Dec. 7  12:0013:00 (Goren) 14:0016:00 (Greenberg)  
Usual office hours of Nicole and Greenberg are cancelled. 
Assignment 10: Submit
by Wednesday Dec. 6, 12:00.
Final Exam:
Friday Dec. 8, 14:0017:00 RUTHERFORD PHYSICS BUILDING.
Tutorial hours:
Day  Time  Room  TA  
Monday  12:3013:30  BURN 1B39  Nicole, M.H.  nicole@math.mcgill.ca
Office: 1132 Office hours: 13:00  15:00 Tuesday at Burnside 9th floor. 
Tuesday  15:00  16:00  BURN 1214  Greenberg, M.  greenberg@math.mcgill.ca, Office: 1036
Office hours: 10:00  12:00 Thursday at Burnside 9th floor. 
Detailed Syllabus
Week  Material  Assignment  Comments  Learn More! (No... it's not on the final!)  
Sept. 6  8  Overview of the course.
Sets and functions. Proofs: idea and techniques. (Induction, negation) 
Non  Recommended reading: Appendices A, B, C in Hungerford.  On AlKhwarizmi.  
Sept. 1115  Proofs: Continued.
Sets and functions; Continued. Number systems: integer, rational, real and complex numbers. Fields and Rings. 
Assignment 1: ps or pdf or dvi  Appendices and Chapter 3.1.  *  
Sept. 1822  Arithmetic in Z: Divisiblity, gcd's and the Euclidean algorithm.  Assignment 2: ps or pdf or dvi  Recommended reading: Chapter 3.1., Chapter 1.
In Question A.4.(b) prove all axioms hold except existence of 1. Note that in Hungerford a ring need not have 1. We (and many other text books) required the existence of 1 because in all cases of interest to us the rings would have 1. 
On Euclid.  
Sept. 2529  Arithmetic in Z: Prime numbers and The Fundamental Theorem of
Arithmetic.
Congruences and modular arithmetic. 
Assignment 3: ps or pdf or dvi  Recommended reading: Chapter 1, Chapter3.13.2. Appendix D. Chapter 2.1.  On prime Numbers. On Pythagoras. On primality testing.  
Oct. 26  Congruences and modular arithmetic.
Finite fields and Fermat's theorem. Primality testing. RSA method. Rings of polynomials: Polynomials, roots, reducible and irreducible elements. The Euclidean algorithm for polynomial rings. 
Example of RSA given in class: ps or pdf or dvi.  Recommended reading: Chapter 2, Chapter 12, Chapter 4.14.2.  On the RSA method.
561, 1105, 1729, 2465, 2821 are Carmichael numbers. Read on Fermat  "the greatest amateur". 

Oct. 913 

* 
Oct. 9 is Thanksgiving (no class). 
*  
Oct. 1620  M Review.
WF Rings of polynomials. 
* 
Oct. 16  Midterm in the evening!
Morning class  end of review! Midterm: psdvipdf . Solutions: psdvipdf . Reading: Chapter 4.34.5. 
David Hilbert. Hilbert's Problems.  
Oct. 2327  Theory of rings: Rings of matrices. Ideals, homomorphisms and quotient rings.  Assignment 5: ps
or pdf
or dvi
.
Part 3 of Question 6 is optional (you don't have to do it). 
Reading: Chapters 3 and 6.  *  
Oct. 30Nov. 3  Modular arithmetic revisited.
Quotients of polynomial rings. 
Assignment 6: ps or pdf or dvi .  Reading: Chapters 3 and 6.  *  
Nov. 610  The first isomorphism theorem for rings.
Maximal and primes ideals; construction of fields. 
Assignment 7: ps or pdf or dvi .  Reading: Chapters 3 and 6.  *  
Nov. 1317  Review of Rings.
Groups: definition and examples (including the symmetric groups and symmetry groups); first properties; subgroups. 
D_6
transparency.
Assignment 8: ps or pdf or dvi . 
Chapter 7, Sections 7.1, 7.2, 7.3.  Evariste
Galois.
The development of Group theory. 

Nov. 2024  Subgroups, cosets, indices and Lagrange's theorem.
Group actions, orbits and stabilizers. 
Assignment 9: ps or pdf or dvi .  Chapter 7, Sections 7.5, 7.9.  Lagrange.  
Nov. 27Dec. 1  CauchyFrobenius formula, applications to combinatorics.
Normal subgroups. Homomorphisms and isomorphisms, the first isomorphism theorem. 
Assignment10: ps
or pdf
or dvi
.
TO BE SUBMITTED DEC. 6, 12:00. 
Notes on Group actions
Chapter 7, Section 7.4. 
Cauchy.
Frobenius. 

Dec. 46  Review  Practice Final  *  * 