AlgebraI

Honours Algebra 2, MATH 251 page last update: April 5, 2008.

Instructor: Eyal Goren
Time and location: MWF 8:35-9:25 in the basement of Burnside Building 1B23.
Office hours: Monday 13:30 - 14:30, BURN 1108.

TA: Gabriel Chenevert
Tutorial Hours: Wednesday 16:30 - 17:30 in BURN 1205
TA office hours: Friday 13:00 - 14:00
MATH help desk: BURN 911. Schedule is on the door.

Syllabus (Calendar description): Linear equations over a field. Introduction to vector spaces. Linear maps and their matrix representation. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Real and complex inner product spaces. Diagonalization of self-adjoint operators.
Prerequisites: MATH 235 or permission of the Department
Text book: The course notes, available for free from this website, are the official textbook for this course. This is the final version. Errata (please bring such to my attention) will be posted here. The course material is everything covered in the notes. Some topics will be discussed very briefly in class and you need to study the notes for a more thorough discussion.
Additional text books: You may want to consult additional texts, but be mindful that notation and definitions may be slightly different. Some texts are the following (both on reserve in the library and the library contains many others, not necessarily inferior, textbooks):

Schaum's Outline of Linear Algebra by Seymour Lipschutz and Marc Lipson (3^rd edition). (You may find this interesting)
Linear algebra by Kenneth Hoffman and Ray Kunze.

Evaluation method:

15% Assignments.(I expect you to do routine easy exercises yourself. The assigenments will contain more challenging problems.)
30% Quizzes. There are going to be 3 quizzes during the term. Each will contain multiple choice questions and one proof question. Each quiz is 90 minutes long and will be held in the afternoon or evening. See dates below. The material for Quiz 1 is everything in the table below, until it says (Quiz 1). For Quiz 2, it is the material listed between (Quiz 1) and (Quiz 2), etc.
55% Final exam.
Calculation of final grade: The better of the method described above and the grade of the final exam alone. Same rule applies to deferred/supplemental exams. Based on past experience, I strongly recommend the strategy of getting a high grade on the assignments and quizzes, instead of gambling on just writing a good final. Students will NOT have the option of make-up/additional work to improve their grades.

Academic integrity: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

Detailed Syllabus *
Date	Material	Assignment	Misc.
January 4	Introduction and motivation.
January 7 - 12	Defn of v. space and subspace. Examples. Sum and intersection, direct sum. Span. Linear dependence.	Assignment 1 Solutions
January 14 - 18	Basis and dimension. Coordinates and change of basis. Linear maps: definitions and first examples.	Assignment 2 Solutions
January 21 - 25	Dim(V) = dim(Ker) + dim(Im). Applications. Quotient spaces. Direct sums. Nipotent operators and projections. (Quiz 1)	Assignment 3 Solutions
January 28 - February 1	Linear maps and matrices. Signs of permuations. The existence of determinant.	Assignment 4 Solutions	Quiz 1: Jan 31, 16:00 - 17:30, MAASS 217 RESULTS SOLUTIONS
Feburary 4 - February 8	Uniqueness and multiplicativity of determinants. Geomteric interpretation of determinants. Laplace's formulas and the adjoint matrix. Systems of linear equations.	Assignment 5 Solutions
February 11 - 15	Row and column space and rank. Two matrices in REF with same row-space are equal. Rank_r(A) = Rank_c(A). Calculating the inverse matrix. (Quiz 2)	Assignment 6 Solutions	There are some typos in the matrix in REF in question (1). (Some zeros need not be zeros..) You all know how it's supposed to look like!
February 18 - 22	The dual vector space. Inner product spaces. Cauchy-Schwartz inequality.		Quiz 2: Feb 21, 16:00 - 17:30, RPHYS 118 RESULTS SOLUTIONS
February 25 - 29	Study Break	Study Break	Study Break
March 3 - 7	Gram-Schmidt. Orthogonal projection. Eigenvalues and eigenvectors.	Assignment 7 Solutions
March 10 - 14	The characteristic polynomial. Diagonalization. The characteristic polynomial. Arithmetic and geometric multiplicities.	Assignment 8 Solutions
March 17 - 19	The minimal polynomial. Cayley-Hamilton. (Quiz 3) The primary decomposition theorem. Diagonalization again. Examples.	Assignment 9 Solutions	Quiz 3: March 20, 16:30 - 18:00, MAASS 217 RESULTS SOLUTIONS typo: Exer. 3 refers to exer. 2 (and not 1).
March 26 - 28	The Jordan canonical form.	Assignment 10 Solutions
March 31 - April 4	Symmetric and self adjoint operators. Applications: the Principal Axis Theorem, Inner products and more.
April 7- April 11	Normal operators. Bilinear forms.

ExampleA
ExampleB
ExampleB1
ExampleC