Reading for first week:

Appendices A-D (sets, functions, fields, complex numbers).
Appendix E (polynomials) - up to and including Corollary 2.
Notes on fields and complex numbers, and notes on finite fields (see above).

Material covered in class: Tuesday Sept 11

Definition of complex numbers, addition and multiplication in the complex numbers.
Finding multiplicative inverses in the complex numbers.
Complex conjugates, absolute value of a complex number.
Abstract definition of a field.

Material covered in class: Thursday Sept 13

Proving some facts about fields using axioms.
Short discussion of examples: determining whether a set with a given addition and multiplication is a field.

Reading for second week:

Notes on integers mod p; DeMoivre's formula (in notes on fields).
Sections 1.2 and 1.3 of the textbook.

Material covered in class: Tuesday Sept 18

Integers mod p (p a prime number); addition and multiplication in the integers mod p.
Proving that the set of integers mod p is a field.
Finding zeros of polynomials over the integers mod p.
Algebraic closure property of the complex numbers.

Material covered in class: Thursday Sept 20

DeMoivre's formula (nth roots of complex numbers)
Definition of vector space, some examples of vector spaces

Reading for week of September 24th:

Sections 1.2, 1.3, 1.4 and 1.5 of the textbook.

Material covered in class: Tuesday Sept 25

Examples of vector spaces, including polynomial vector spaces;
Definition of subspace of a vector space;
Determining whether a nonempty subset W of a vector space V is a subspace of V.

Material covered in class: Thursday Sept 27

Definition of the vector space of m by n matrices over a field F.
Showing that the intersection of subspaces is a subspace.
Definition of linear combination and span(S) (S a subset of V).
Showing that span(S) is a subspace of V.

Material covered in class: Tuesday October 2

Examples of spans of subsets in various vector spaces.
Showing that if S is a subset of a subspace W of V, then span(S) is a subset (in fact a subspace) of W.
Definition of linearly dependent and linearly independent sets.
A nonempty set S is linearly independent if and only if each element of span(S) can be expressed uniquely as a linear combination of distinct vectors from S.
A set containing two distinct vectors is linearly dependent if and only if the first vector is a scalar multiple of the second vector.

Material covered in class: Thursday October 4

Definitions of finite-dimensional vector space, basis of vector space.
Example: Let V be the vector space of polynomials over the complex numbers of degree at most 4.
Find a basis of the subspace W = { f(x) in V | f(i)=0 }.