| - Integral elements and integral closure.
- Localization and local properties. - Dedekind domains. - Valuations and completions. - Decomposition and ramification. - Trace and norm. |
- Discriminant and different.
- Inertia and decomposition groups. - Class groups and Minkowski's bound. - Units and Dirichlet's theorem. - Fields of low degree. - Cyclotomic fields. |
Lecturer: Eyal Z. Goren Office:
BURN1108
Tel: 398-3815
Office Hours: Mon 10:30-11:30,
Wed 10:30-11:30 and 16-17.
Special Office Hours for April 11-30:
(Usual office
hours are cancelled)
Goren:
April 24 8:30-10:30, April 25 8:30-10:30.
D'amours:
April 28 14:00 - 16:00, April 29 14:00 - 16:00.
(plus you can
try and catch D'amours any other time for questions).
You can pick
up assignments 10, 11 and old assignments too from D'amours after April
15.
TA: Martin D'amours. Office:
BURN 1036 Office hours: Mon 15:30-16:30,
Fri 1:30-2:30.
Tutorial hours: Mon 14:35 -15:25
ARTS W20, Wed 17:00-18:00, BURN 920. (Note: those are two sessions, covering
the same material, that are optional though highly recommended.)
Marker: Thomas Cocolios.
Time of Course: MWF 8:30-9:30 Lecture
Hall: Burnside Hall 1B24
Dates of exams:
Midterm: Wednesday, March 5, 8:35-9:25.
Final: Wednesday April 30
- 14:00 GYM.
| Structure: |
|
| Material: |
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| 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: The grade on the assignments 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. | ||
Syllabus: Vector spaces, spanning
sets, basis, dimension. 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
self-adjoint 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).
The main challenges of the course. (1)
Writing
rigorous proofs. There is a strong emphasis in this course on complete
proofs of almost all result. The language is very formal and precise. Students
are required to be able to right rigorous proofs.
Recommendation:
mimic the style of the lecturer (i.e. me), at least initially. A proof
should state clearly the given facts and the theorems assumed known and
proceed from there by logical deductions, that are clearly stated,
to the conclusion one seeks. (2) The concepts of a vector
space, basis, dimension and of a linear transformation. The challenge
is in digesting the abstract concepts of vector spaces and linear transformations
in a "coordinate free" approach. Recommendation: this requires
deep thinking and having many examples at your fingertips. One should definitely
allow time just to digest the concepts and intuition will be created on
the basis of many examples. (3) Applying the abstract theory
in concrete problems. The syllabus compels us to spend most of the
lectures on definitions, theorems and proofs. We shall try, of course,
and supply examples and applications in class. However, most of the applications
and problems requiring calculations will be given in the assignments. The
challenge would be to see the applicability of the theorems we prove in
class to concrete situations. Recommendation: come to tutorials
and office hours. Attempt additional questions in the text book (which
contains solutions).
A general recommendation. Study
with other people, preferably neither stronger nor weaker than you. However,
in the end of the day, each person has to right his or her own solutions
to the assignments.
Detailed Syllabus
| Week | Material | Assignment | Comments and Corrections | Learn More! (No... it's not on the final!) |
| January 6-10 | Motivation for studying vector spaces, linear map and matrices. Defn of a v.sp., subspace and examples. Sum and direct sum of subspaces. Linear dependence, span and spanning set. | Assign. 1 pdf
Assign. 1 dvi |
Use textbook for more examples and careful verification of the axioms
for them.
Minor Corrections: min{d(x; 0) : x e C; x ¹ 0} and A linear code C corrects t errors if and only if the Hamming distance of every two distinct elements of C is at least 2t + 1. |
History
of abstract linear spaces.
(linear spaces = vector spaces). |
| January 13-17 | Steinitz's substitution lemma, basis, dimension, dim (W+U). Basis, coordinates, change of basis. | Assign. 2 pdf
Assign. 2 dvi |
||
| January 20-24 | Linear maps: def., Ker and Im, examples. Non-singular maps, isomorphism, composition. dim(Ker)+dim(Im) = dim(V). dim(U+W) - another proof. Nilpotent operators.Fitting's lemma. | Assign. 3 pdf
Assign. 3 dvi |
||
| January 27-31 | Projections. Linear transformations and matrices. Examples. Determinants. | Assign. 4 pdf
Assign. 4 dvi |
In question 7 read 'permutation' instead of 'transformation' | hand out on permuations |
| February 3-7 | Determinants. Existence and Uniqueness, multiplicativity, Laplace's Theorem, the adjoint matrix. Linear equations. | Assign. 5 pdf
Assign. 5 dvi |
||
| February 10-14 | Linear equations: row space and column space; reduced echelon forms; row-rank = column rank; Cramer's rule. | Assign. 6 pdf
Assign. 6 dvi |
||
| February 17-21 | Inner product spaces; metric and norm; Cauchy-Schwartz inequality; the triangle inequality; positive definite Hermitian matrices; orthonormal bases and Gram-Schmidt process; orthogonal projection; the method of least squares. | Assign. 7 pdf
Assign. 7 dvi |
There is a typo in the deadline for submitting. You can submit ass. 7 until March 7, 12:00. | |
| February 24-28 | STUDY BREAK | STUDY BREAK | STUDY BREAK | STUDY BREAK |
| March 3-7 | Eigenvalues and eigenspaces. The characteristic polynomial. Algebraic and geometric multiplicity. | MIDTERM ON MARCH 5
Everyone should be sitted by 8:30. SPECIAL T.A. OFFICE HOUR: TUESDAY, MARCH 4, 10:30-11:30. Solutions to Midterm: dvi or pdf |
The midterm has 3 questions. One is to prove a theorem proved in class; the other is to prove a couple of easy propositions you didn't see before; the last is computational question of the kind appearing in the assignments. | |
| March 10-14 | Diagonalization. Application to recursion sequences. Criterion: m_g = m_a for all e.values; Cayley-Hamilton; the minimal poly. and the char. polyl. Diagonalization and the min. poly. m_A|D_A|m_A^n | Assign. 8 pdf
Assign. 8 dvi |
||
| March 17-21 | The Primary Decomposition Theorem. Diag'able iff m_A is a product of distinct lin. factors. Jordan canonical form. | Assign. 9 pdf
Assign. 9 dvi |
Example given in class
In Q. 3, Assign. 9, assume that the char. of the field is not 2 (so every non-zero scalar is has two distinct roots). |
|
| March 24-28 | Jordan canonical form (cont'd). Self-adjoint operators. | Assign. 10 pdf
Assign. 10 dvi |
Example given in class | |
| March 31- April 4 | Self-adjoint operators (cont'd). Application to inner products. Normal operators and the Spectral Theorem. | Assign. 11 pdf
Assign. 11 dvi |
Another example of JCF | |
| April 7 - 11 | The dual space. Bilinear forms. | |||
| Final Exam Solutions (Sketch) |