** Instructor: Fiona Murnaghan **

** Class location: ** Tuesday MP103 (at 1:10pm); Thursday MP202 (at 1:10pm)

** Office hours before exam: ** Friday Dec 5 and Monday Dec 8 at 2pm.

Review questions for term test (solutions not provided)

** Office Hours: **

** Tutorial information:** Tutorials start Thursday Sept 18 (2:10-4pm)

Notes on bases and dimension (Section 1.6 of text).

Note that late problem sets will not be accepted, except in extreme situations (such as serious illness or hospitalization).

Problem set 1 (due Thursday September 25th).

Problem set 2 (due Thursday October 2nd).

Problem set 3 (due Thursday October 9th).

Problem set 4 (due Thursday October 16th).

Problem set 5 (due Thursday November 6th).

Problem set 6 (due Thursday November 13th).

Problem set 7 (due Thursday November 20th).

Problem set 8 (not to be handed in).

Problem set 1 partial solutions.

Problem set 2 partial solutions.

Problem set 3 partial solutions.

Problem set 4 partial solutions.

Problem set 5 partial solutions.

Problem set 6 partial solutions.

Problem set 7 partial solutions.

Problem set 8 partial solutions.

Students who are interested in the Putnam Competition, or other math competitions: please contact Donna Birch at dbirch@math.toronto.edu.

**Final exam:** the date and time are set by the Faculty of
Arts and Science - please refer to the December exam schedule.

Reading for the first week: notes on fields and notes on integers modulo p; Appendices A-D of text (pages 549-561); Appendix E of text, pages 562-564, up to Corollary 2 on page 564.

Material covered on Tuesday September 9th: complex numbers; fields and field axioms (see notes on fields); proofs using field axioms.

Material covered on Thursday September 11th: some proofs using field axioms; complex conjugates, polar coordinates, DeMoivre's formula.

Reading for the second week: notes on integers mod p; Sections 1.1 and 1.2 of the text.

Material covered on Tuesday Sept 16th: Example: find all complex numbers whose cubes are equal to -3i. (There are 3). Examples: demonstrating whether a given set F is a field, or verifying that a field axiom fails. Definition of addition mod p and multiplication mod p. Verifying some field axioms for integers mod p.

Material covered on Thursday Sept 18th: Existence of multiplicative inverses in integers mod p; Examples involving integers mod 7. No time to start vector spaces.

Reading for the third week: Sections 1.1--1.4 of the text.

Material covered on Tuesday Sept 23rd: Definition of vector space; examples; definition of subspace; statement of Theorem 1.3(subspace test).

Material covered on Thursday Sept 25th: Examples using the subspace test; proof of Theorem 1.3(subspace test).

Reading for the week Sept 29-Oct 3: Sections 1.4 and 1.5.

Material covered on Tuesday Sept 30th: Linear combinations and span; linear independence and dependence.

Material covered on Thursday Oct 2nd: Examples involving linear independence; example:basis of F^n; Definition of basis for a vector space V.

Reading for the week Oct 6-10: Section 1.6 and notes on bases and dimension (posted above).

Material covered on Tuesday Oct 7th: Bases and dimension; examples.

Material covered on Thursday Oct 9th: Bases and dimension; example- basis of P_n(F) consisting of polynomials of degree n; dimension of a subspace (Theorem 1.11 of text).

Reading for the week Oct 14--17: Examples on pp.50-51; Sections 2.1 and 2.2.

Material covered on Tuesday Oct 14th:

Material covered on Thursday Oct 16th:

Reading for the week Oct 20--24: Sections 2.2 and 2.3 of text.

Material covered on Tuesday Oct 21st: Theorem 2.3, Theorem 2.5, Theorem 2.6, related results and examples. Definition of ordered basis and coordinate vectors.

Material covered on Thursday Oct 23rd:

Reading for week October 27-31: Sections 2.2-2.4 of text.

Material covered on Tuesday October 28th:

Material covered on Thursday October 30th:

Reading for the week Nov 3-7: Properties of matrices from Section 2.3, Sections 2.4 and 2.5.

Material covered on Tuesday Nov 4th:

Material covered on Thursday Nov 6th:

Reading for the week Nov 10-14: Sections 2.5, 3.1 and 3.2.

Material covered on Tuesday Nov 11th:

Material covered on Thursday Nov 13th:

Reading for the rest of the course:

Material covered on Tuesday Nov 18:

Material covered on Thursday Nov 20:

Material covered in Tuesday November 25th:

Note: The material in Section 5.2 on systems of differential equations and direct sums will not be covered and will not be on the final exam.

Material covered on Thursday Nov 27:

Material covered on Tuesday Dec 2: