Philosophy
This class is reading-and-writing 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 n-forms 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 Cayley-Hamilton'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 Gram-Schmidt'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 self-adjoint 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)