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):

Evaluation method:
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 *



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

January 14 - 18
Basis and dimension. Coordinates and change of basis. Linear maps: definitions and first examples.
Assignment 2

January 21 - 25
Dim(V) = dim(Ker) + dim(Im). Applications. Quotient spaces.  Direct sums. Nipotent operators and projections. (Quiz 1) Assignment 3

January 28 - February 1
Linear maps and matrices. Signs of permuations. The existence of determinant.
Assignment 4

Quiz 1: Jan 31, 16:00 - 17:30, MAASS 217


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

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
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


 February 25 - 29
Study Break Study Break Study Break
March 3 - 7
Gram-Schmidt. Orthogonal projection. Eigenvalues and eigenvectors.
Assignment 7

March 10 - 14
The characteristic polynomial. Diagonalization. The characteristic polynomial. Arithmetic and geometric multiplicities. 
Assignment 8

March 17 - 19
The minimal polynomial. Cayley-Hamilton. (Quiz 3) The primary decomposition theorem. Diagonalization again. Examples.  Assignment 9
Quiz 3: March 20, 16:30 - 18:00, MAASS 217

typo: Exer. 3 refers to exer. 2 (and not 1).
March 26 - 28
The Jordan canonical form.
Assignment 10

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.