MAT 240F - Algebra I

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:

Bahen 6266
Tuesday 5:10-6pm, Wednesday 5-5:30pm, and by appointment.

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

Last name starting with a letter between A and K: SS2108; TA: Nevena Francetic
Last name starting with a letter between L and Z: MP202; TA: Jordan Watts

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:

Example-finding a basis for a subspace;
Corollary on page 51 of text;
Definition of linear transformation; examples;
Proof of property 4 on page 65.
Null space N(T) of a linear transformation T: V --> W; N(T) is a subspace of V;
Null space N(T) of T is zero if and only if T is one-to-one.

Material covered on Thursday Oct 16th:

nullity, range, and rank of a linear transformation T :V --> W ;
The range R(T) of T is a subspace of W;
If span(S)=V, then R(T)=span(T(S)), where T(S)={ T(x) | x in S }.
If dim(V) is finite, then dim(V)=nullity(T) + rank(T) (Dimension Theorem).

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:

The coordinate function relative to an ordered basis is a bijective linear transformation (Theorem 2.21).
Recall definition of vector space of m by n matrices over a field F
;
Definition of matrix of a linear transformation T:V-->W relative to ordered bases for V and W.
Example of computing the matrix of a linear transformation.
Statement of Theorem 2.14: The cooordinate vector of T(x) relative to the ordered basis for W is equal to the product of the matrix for T times the coordinate vector of x relative to the ordered basis for V.

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

Material covered on Tuesday October 28th:

Example given a matrix A having a particular form. If A is the matrix of T relative to some bases, deduce properties of T from properties of A.
Statement and proof of Theorem 2.14 (proof not exactly as in text).
Definition of L(V,W); L(V,W) is a vector space;
Definition of isomorphism; Statement and proof of Theorem 2.20.

Material covered on Thursday October 30th:

Example: Proving existence of a linear transformation T with certain properties.
Definition of composite of two linear transformations.
Statement of Theorem 2.11.

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:

Proof of Theorem 2.11.
Statement of Theorem 2.12 and 2.16.
Definition of invertible linear transformation and invertible matrix.
If T is invertible then the inverse of T is linear. (Theorem 2.17)
Theorems 2.18 and 2.19.
Example: A proper subspace of P(F) that is isomorphic to P(F).
Formula for the inverse of an invertible 2 by 2 matrix.

Material covered on Thursday Nov 6th:

If T is in L(V,W) and T is one-to-one, then a subset S of V is linearly independent if and only if T(S) is linearly independent.
If T is in L(V,W) and T is invertible, then S is a basis for V if and only if T(S) is a basis for V.
If T is in L(V,W), dim(V) is finite, T is invertible, and V' is a subspace of V, then dim(V')=dim(T(V')).
An n by n matrix A is invertible if and only if the columns of A are linearly independent.

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

Material covered on Tuesday Nov 11th:

If A is a matrix, rank(A) = dim(span{columns of A}).
If A is the matrix of a linear transformation T, then rank(A)=rank(T). (Theorem 3.3)
Statement of Theorem 3.6.
Elementary transformations, elementary row operations, reduced row echelon form (p.185).
Computing inverses of matrices using elementary row operations.

Material covered on Thursday Nov 13th:

Example: finding the inverse of T by finding the inverse of the matrix of T.
Outline of proof of Theorem 3.6.

Reading for the rest of the course:

Eigenvalues, eigenvectors and characteristic polynomials- Section 5.1.
Definition of determinant - Section 4.2; Properties of determinants - Section 4.3 (Theorem 4.7 and Corollary).
Section 2.5 - change of coordinate matrix and similarity of matrices.
Diagonalizability: Section 5.2.
The Cayley-Hamilton Theorem - Theorem 5.23 (will not be on final exam).

Material covered on Tuesday Nov 18:

Definition of eigenvalue, eigenvector, eigenspace, diagonalizable.
Some examples with dim(V)=2: finding eigenvalues and eigenvectors.
A scalar c in F is an eigenvalue of a linear transformation T from V to V if and only if T-cI is not one-to-one.
Example: finding eigenvalues using determinants, dim(V)=4.

Material covered on Thursday Nov 20:

Returning to example from last class: Finding a basis for the eigenspace corresponding to the eigenvalue 3.
For the same example, showing that T is not diagonalizable.
Change of coordinates: Theorem 2.23.
Statement of Theorem 4.7, and the corollary on page 223.
Definition of the characteristic polynomial of T (V finite-dimensional).
Showing that the characteristic polynomial of T is independent of the basis used to compute the matrix of T.

Material covered in Tuesday November 25th:

Example: If T is diagonalizable, and a, b are in F, show aTT + bT is diagonalizable.
Example: Specific T such that TT-2T is diagonalizable but T is not diagonalizable.
Proposition: A set x_1,...,x_m of eigenvectors of T corresponding to distinct eigenvalues is linearly independent.
Corollary: If T has n=dim V distinct eigenvalues, then T is diagonalizable.
Theorem 5.8.
Corollary: The dimension of the subspace spanned by all eigenvectors of T is equal to the sum of the dimensions of the eigenspaces of T.
Theorem 5.9. (See also page 269 - comments on testing for diagonalizability.)

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:

Using determinants to prove Theorem 5.7.
Example: Solving question 8 from Problem Set 8.

Material covered on Tuesday Dec 2:

General comments about properties of determinants.
Example: A linear transformation T with every real number as an eigenvalue.
Statement of the Cayley-Hamilton Theorem (not material for exam).
Proof of the Cayley-Hamilton Theorem for T diagonalizable.