Office hours week of Dec 10-14: Tuesday 3:10-4; Thursday 3:10-4.
Final exam: the date and time are set by the Faculty of
Arts and Science - please refer to the December exam schedule.
Information about the final exam.
Note that late problem sets will not be accepted, except
in extreme situations (such as serious illness or hospitalization).
Information about term test marks.
Review questions for term test.
Problem set 1 (due Thursday September 27th)
Problem set 2 (due Thursday October 4th)
Problem set 3 (due Thursday October 11th)
Problem set 4 (due Thursday October 18th)
Problem set 5 (due Thursday November 8th)
Problem set 6 (due Thursday November 15th)
Problem set 7 (due Thursday November 22nd)
Problem set 8 (due Thursday November 29th)
Problem set 9 - will not be handed in,
but will cover material on the final exam.
Page 1 of Problem set 1 partial solutions
Page 2 of Problem set 1 partial solutions
Page 3 of Problem set 1 partial solutions
Problem set 2 partial solutions
Page 1 of Problem set 3 partial solutions
Page 2 of Problem set 3 partial solutions
Page 3 of Problem set 3 partial solutions
Problem set 4 partial solutions
Term test solutions
Page 1 of Problem set 5 partial solutions
Page 2 of Problem set 5 partial solutions
Page 3 of Problem set 5 partial solutions
Problem set 6 partial solutions
Page 1 of Problem set 7 partial solutions
Page 2 of Problem set 7 partial solutions
Page 3 of Problem set 7 partial solutions
Problem set 8 partial solutions
Problem set 9 partial solutions to some questions.
Course information and marking scheme
Notes on fields, complex numbers, and DeMoivre's
formula.
Notes on finite fields (integers modulo p, p prime).
Reading and material covered in the
first four weeks (Sept 10-Oct 5).
Reading for the week of October 9th:
Section 1.6; notes on bases and dimension; systems of equations(Section 1.4).
Notes on bases and dimension
Material covered in class: Tuesday Oct 9
Finished example from Oct 4.
Lemmas 1 and 2, Theorems 1, 2 and 4, Corollary 1 from notes on bases
and dimension.
Example: Let V=P_4(C) and let S={ ix^4+x^3,x^4+x^2,-x^4+i,-ix^4+x}.
Find a basis of V that contains S.
Example: Let S = { (1,1,1,-1), (2,0,-1,0)} in the
vector space V of 4-tuples of real numbers.
Find a basis of V that contains S.
Material covered in class: Thursday Oct 11
Example 1: Finding a basis of P_n(F) that consists
entirely of polynomials of degree n.
Example 2: Showing that there does not exist a
basis of P_n(F) that contains only polynomials with f(0)=0.
Example 3: Finding a basis of 2 x 2 matrices over F
such that the first entry of the first column of each basis vector
is equal to 1.
If W is a subspace of a finite-dimensional vector space V, then
W is finite-dimensional and dim(W) is less than or equal to dim(V).
If W is as above, each basis of W is contained in some basis of V.
Reading for the week of October 15th:
Sections 1.6, 2.1 and 2.2.
Material covered in class: Tuesday Oct 16
Proving that any two bases in a finite-dimensional vector space contain the same number of vectors.
Several examples of finding a basis of span(S), S a subset of V.
Material covered in class: Thursday Oct 18
Definition and examples of linear transformations.
Material covered in class: Tuesday Oct 23
Property 4 of linear transformations (as on page 65 of text).
Theorem 2.6 and Corollary (see text).
Null space, nullity, range and rank of a linear transformation.
Theorem 2.1 + Nullity of T is zero if and only if T is one-to-one.
Theorem: If span(S)=V, then span(T(S))=R(T), where T(S) is the
set of all T(x), as x ranges over S.
Stated dimension theorem (Theorem 2.3 in text).
Used dimension theorem to prove Theorem 2.5.
If dim(V)=m, dim(W)=n and m is less than n, then T is not onto.
If dim(V)=m, dim(W)=n and m is greater than n, then T is not one-to-one.
Example V=P(F), T(f)(x)=xf(x), T is linear, one-to-one, not onto.
Material covered in class: Thursday Oct 25
Proof of the dimension theorem.
Example: finding a basis for N(T) and R(T).
Material covered in class: Tuesday Oct 30
Ordered bases, coordinate functions, matrices of linear transformations.
The coordinate function is a bijective linear transformation
(Theorem 2.21, section 2.4).
Example of finding the matrix of a particular linear transformation.
Definition of matrix multiplication (page 87).
Theorem 2.14 (section 2.3).
Material covered in class: Thursday Nov 1
Definition of the "left-multipication transformation" (page 92).
Most parts of Theorem 2.15.
First two parts of Theorem 2.12.
Definition of L(V,W); L(V,W) is a vector space (Theorem 2.7).
Theorem 2.8 (section 2.2); statement of Theorem 2.20 (section 2.4).
Material covered in class: Tuesday Nov 6
Finished proof of Theorem 2.20.
Definition of isomorphism; corollary on page 103 of text.
Composition of linear transformations; examples.
Theorem 2.11 (matrices of compositions as products of matrices).
Material covered in class: Thursday Nov 8
Showed the the inverse of a bijective linear transformation is linear.
Theorem 2.19.
Theorem 2.18.
Material covered in class: Tuesday Nov 13
Example: constructing an isomorphism satisfying certain properties.
Theorem: If S is a basis of V and T is in L(V,W), then T is invertible
if and only if T(S) is a basis of W.
Change of coordinate matrices (section 2.5), including one example.
Formula for the inverse of a 2 by 2 invertible matrix.
Similarity of matrices.
Material covered in class: Thursday Nov 15
Elementary row and column operations; elementary matrices.
Theorem 3.1; Theorem 3.2.
Definition of rank of a matrix; statement of Theorem 3.6.
Material covered in class: Tuesday Nov 20
Theorems 3.3, 3.4 and 3.7; example related to Theorem 3.6.
Finding matrix inverses using elementary row operations.
Material covered in class: Thursday Nov 22
Definition of determinant of an n by n matrix.
Properties of determinants.
A and B n by n matrices: det(AB)=det(A)det(B) (proved when A is invertible).
If A is invertible, det(A) is not zero.
Material covered in class: Tuesday Nov 26
If A is not invertible, then det(A)=0.
det(AB)=det(A)det(B) (proof in the case A not invertible).
det(transpose of A)=det(A).
Definition of eigenvalue, eigenvector, eigenspace,
T diagonalizable.
A couple of examples of finding eigenvalues and eigenvectors.
Material covered in class: Thursday Nov 28
Two examples of finding characteristic polynomials,
eigenvalues and eigenvectors, testing for diagonalizability.
Theorem: Eigenvectors corresponding to distinct eigenvalues
are linearly independent (Theorem 5.5).
Theorem: If dim(V)=n and T has n distinct eigenvalues, then T is
diagonalizable.
Material covered in class: Tuesday Dec 4
Basic ideas of proofs of theorems from last time.
Theorems 5.7, 5.8 and 5.9.
More examples.
Note: The part of section 5.2 on direct sums will not be covered.
Thursday Dec 6
Examples from Problem set 9.
Other examples?
If time permits, a brief summary of
systems of linear equations (sections 3.3 and 3.4). (not required for exam)
Note: Sections 2.6 and 2.7 will not be covered in Mat 240.
