This is
the website for MATH 672, Vector spaces, for the Fall 2013
semester. Scroll down for a continuously updated schedule.
Important data to keep in mind:
Important data to keep in mind:
 Lectures are MWF 1:25pm2:15pm, at Purnell Hall 324B
 Instructor: FranciscoJavier Sayas (here's my website)
 Office
hours: MW after class, and by appointment. (Please,
avoid asking questions by email.)
 Textbook. A (Terse)
Introduction to Linear Algebra, by Katznelson &
Katznelson. (AMS). We will cover Chapters 1 through 7.
Philosophy
This class is readingandwriting intensive. You'll be asked to read carefully different sections of the book. In class, we'll be working on problems and clarifying concepts, but you will not be getting clean classnotes to work with. (The textbook is where to study from.)
Precision and full rigor in presentation of the proofs is expected of all the students. Most of the problems will be about concepts and proofs, and there'll be very few exercises carrying a lot of computation. For training purposes, you'll be asked to memorize one particular item (most often a proof) every now and then.
Some additional materials can be found in the materials page of this website.
The schedule
Check often for updates, links and what not. Exercises and concepts marked in italics are more difficult. Take special care to understand the difficult concepts.
This class is readingandwriting intensive. You'll be asked to read carefully different sections of the book. In class, we'll be working on problems and clarifying concepts, but you will not be getting clean classnotes to work with. (The textbook is where to study from.)
Precision and full rigor in presentation of the proofs is expected of all the students. Most of the problems will be about concepts and proofs, and there'll be very few exercises carrying a lot of computation. For training purposes, you'll be asked to memorize one particular item (most often a proof) every now and then.
Some additional materials can be found in the materials page of this website.
The schedule
Check often for updates, links and what not. Exercises and concepts marked in italics are more difficult. Take special care to understand the difficult concepts.
Week 
Lecture 
Section 
Description
and problems 

1 
08/28 
W 
(So you
know about linear systems, don't you?) Slides 

08/30 
F 
1.1 
Groups
and fields Ex 1.1.2, 1.1.3, 1.1.4 

2 
09/04 
W 
1.2 
Vector
spaces (definition, examples, and basic
properties) 
09/06 
F 
Isomorphisms
and subspaces 

3 
09/09 
M 
1.2 
Direct
sums and
quotient spaces Ex 1.2.1, 1.2.2, 1.2.3, 1.2.4, 1.2.7, 1.2.8, 1.2.9,1.2.10 
09/11 
W 
1.3 
Linear
combinations, linear independence and spanning
sets 

09/13 
F 
Bases
of vector spaces Quiz #1 (solve a linear system, memorize a proof, explain a proof, define a couple of concepts) 

4 
09/16 
M 
1.3 
The
dimension of a finite dimensional space Ex 1.3.2, 1.3.3, 1.3.4, 1.3.5, 1.3.6, 1.3.7, 1.3.10, 1.3.12, 1.3.13 
09/18 
W 
1.4 
Linear
systems (READING SECTION) Ex 1.4.1, 1.4.2, 1.4.3., 1.4.7, 1.4.8, 1.4.9. 

09/20 
F 
2.1 
Linear
operators Quiz #2 (definitions, one proof, check that something is a basis) Ex 2.1.2, 2.1.3, 2.1.4 

5 
09/23 
M 
2.4 
Operators
and matrices 
09/25 
W 
2.2 
Operator
multiplication and the algebra L(V) Ex 2.2.1, 2.2.2., 2.2.4 

09/27 
F 
2.3 
Matrix
multiplication and the algebra of square matrices Quiz #3 (definitions, one proof, the matrix for a linear operator) [All exercises of Section 2.3 are interesting. Try them.] 

6 
09/30 
M 
2.4 
Changes
of basis 
10/02 
W 

10/04 
F 
Similarity
of matrices and operators Ex 2.4.1, 2.4.4, 2.4.5, 2.4.6, 2.4.8 

7 
10/07 
M 
2.5 
Kernel,
range, nullity, and rank Ex 2.5.2, 2.5.3, 2.5.4, 2.5.8, 2.5.9, 2.5.10, 2.5.11, 2.5.12, 2.5.13 
10/09 
W 
3.1 
Linear
functionals Take home part of the midterm due 

10/11 
F 
The
dual basis Ex 3.1.1, 3.1.2, 3.1.3, 3.1.5, 3.1.6a, 3.1.8 Quiz #4 (explain a proof, and two exercises) 

8 
10/14 
M 
FIRST MIDTERM EXAM (Chapters 1 and 2)  
10/16 
W 
3.2  The
adjoint of an operator ... 

10/18 
F 
... and
transposition of matrices Ex 3.2.1, 3.2.2, 3.2.3 

9 
10/21 
M 
4.2 
Bilinear
operators and forms 
10/23 
W 
Multilinear
forms Ex. 4.2.1, 4.2.2, 4.2.3, 4.2.4, 4.2.6, 4.2.7 

10/25 
F 
4.3 
Alternating
nforms Quiz #5 

10 
10/28 
M 
4.4 
The
determinant of an operator Ex 4.4.1, 4.5.2, 4.5.3, 4.5.4, 4.5.6, 4.5.7, 4.5.8 
10/30 
W 
5.1 
Eigenvalues
and the characteristic polynomial Ex 5.1.2, 5.1.3, 5.1.4, 5.1.5, 5.1.6 

11/01 
F 
5.2 
Invariant
subspaces Quiz #6 

11 
11/04 
M 
5.2 
The
minimal polynomial for a vector w.r.t. an operator
and CayleyHamilton's Theorem Ex. 5.2.2, 5.2.3, 5.2.4 
11/06 
W 
5.3 
The
minimal polynomial for an operator Ex 5.3.1, 5.3.3, 5.3.4, 5.3.5, 5.3.6, 5.3.15, 5.3.16, 5.3.17 

11/08 
F 
5.2 
Every
complex matrix is similar to a triangular matrix Quiz #7 

12 
11/11 
M 
Preview
of the Jordan canonical form 

11/13 
W 
6.1 
Inner product spaces  
11/15 
F 
SECOND MIDTERM EXAM (Chapters
1 to 5) (Solutions are available at materials page) 

13 
11/18 
M 
6.1 
Orthogonality Take home part of the midterm due 
11/20 
W 
Orthonormal
bases and GramSchmidt's method Ex. 6.1.3, 6.1.4, 6.1.5, 6.1.8, 6.1.9, 6.1.14, 6.1.15 

11/22 
F 
6.2 
Duality
in inner product spaces Ex. 6.2.1, 6.2.2, 6.2.3, 6.2.4 

14 
11/25 
6.3 
The spectral
theorem for selfadjoint operators Ex 6.3.1, 6.3.5, 6.3.6 

Thanksgiving week 

15 
12/02 
M 
6.4 
Normal operators Quiz #8 due (Solutions are available at materials page) 
12/04 
W 
6.5 
Unitary
operators 

12/09 
FINAL EXAM (COVERS EVERYTHING) 