Students will suffer no academic penalty if they choose not to cross the picket lines or have difficulty reaching the campus.

**Lectures and tutorials will continue as scheduled.** Students will be
expected to keep up with the material covered in the course. Lectures
will follow the text and details of what was in each day's lecure will be
posted on this site.

Tutorial sheets will be posted on this website on a weekly basis.

**Lecture on Friday, Oct. 27:**

The answers to tutorial sheet 5 have been posted on this website. No new tutorial sheets have been handed out.

We covered all of the material in section 4.1 of the text and began section 4.2 (up to but not including Reflection Operators).

**Lecture on Monday, Oct. 30**

We continued our study of linear transformations (section 4.2 of the
text) by looking at the specific types of transformations: reflections,
projections and rotations (so far only in R^{2}). We will
continue with this on Wed.

**Lecture on Wednesday, Nov. 1**

Information on study groups was handed out. If you want to join such a
group send an email to Donna Kotsopoulos

Donna_Kotsopoulos@edu.yorku.ca

and indicate your preferred time: MWF 12:00; MWF 3:00; MWF10:30

We continued to study linear transformations. Topics covered were
rotations in R^{3}, dilations and contractions and the composition
of two transformations. We then began to consider the material in section
4.3 of the text by defining a one-to-one transformation. This will be
continued on Friday.

**Lecture on Friday, Nov. 3**

We finished section 4.3 of the text by studying the inverse of a linear
transformation, the general definition of a linear transformation
(Theorem 4.3.2) and the
construction of the standard matrix for a linear transformation by its
effect on the standard basis of R^{n} (Theorem 4.3.3).

**Lecture on Monday, Nov. 6**

We began the study of general vector spaces by considering real vector spaces (section 5.1 of the text). In particular, we looked at the definition of a vector space and Thm 5.1.1. We also considered several examples of vector spaces, e.g. 2x2 matrices and polynomials of degree 2.

**Lecture on Wednesday, Nov. 8**

We continued our study of vector spaces by defining a subspace of a vector space (section 5.2 of the text). We proved Thm 5.2.1 on the conditions for a subset to be a subspace and considered several examples of subspaces.

**Lecture on Friday, Nov. 10**

We considered two further examples of subspaces - the solutions of a system of linear homogeneous equations and a line through the origin. We then went on to consider linear combinations of vectors and how one could determine the coeffiicients in such a combination (Section 5.2 of the text).

**Lecture on Monday, Nov. 13**

Tutorial sheet 7 was handed out. It is on the web. The answers to
tutorial sheet 6 have been posted as well.

Today we finished section 5.2 by considering the span of a set of vectors
from a vector space. We began section 5.3 by introducing the concept of
linear dependence and independence of vectors.

Note that you are not responsible for any of the examples involving
calculus.

**Lecture on Wednesday, Nov. 15**

The date for Term Test 2 was announced. See the main course page for
details.

Today we considered a number of examples illustrating linear dependence
and independence of vectors and derived some results based on these ideas
(section 5.3 of the text).

We then defined a **basis** for a vector space and did an number of
examples (section 5.4 of the text). Note that while Theorem 5.4.2 is
correct the proof given in the text is not valid for part (b). We used
the form of the proof given to prove Theorem 5.4.3 directly by assuming we
had two bases with different number of vectors and showing this leads to a
contradiction, i.e. that the basis with the larger number of vectors must
be linearly dependent.

**Lecture on Friday, Nov. 17**

We completed the study of the dimension of a vector space by considering a number of examples. We then defined the row space, column space and null space of a matrix as well as the rank and nullity of a matrix (sections 5.5 and 5.6 of the text). Students are responsible for knowing only these definitions and not the theorems or methods of determining these spaces.

**Lecture on Monday,Nov. 20**

We continued our brief look at sections 5.5 and 5.6 of the
text. Students should know the results of Theorem 5.5.2 (but not the
proof) and be able to put the solution of a linear system with an infinite
number of solutions into this form (see example 4 of section
5.5). Students should also know the meaning of rank and nullity from
section 5.6 and the results of Theorem 5.6.3.

We began a brief study of inner product spaces (Chapter 6) by reviewing
the inner product in R^{n} and the concept of length of a vector
and the angle between two vectors. In particular, we are interested in
a set of orthonormal vectors, that is, vectors which are orthogonal to one
another and have length one (section 6.3). We then proved Theorem 6.3.1.

**Lecture on Wednesday, Nov. 22**

Term test 2 was held today. For those students who did not write it, another test will be given in the extended term after the strike has ended.

**Lecture on Friday, Nov. 24**

Term test 3 was announced. See notice on the home page of this
course.

Tutorial sheet 8 was handed out today. It is on this website.

Today we finished off our study of orthonormal basis (section 6.3) by
considering a couple of examples. Then we considered the Gram-Schmidt
method for constructing a set of orthonormal vectors out of a linearly
independent set of vectors and did an example.

**Lecture on Monday, Nov. 27**

The marked tests were returned today. If you did not get yours, they
are available in my office.

We began our in-depth study of eigenvalues and eigenvectors
(Chapter 7 of the text). We
considered a number of examples and discussed the ideas of eigenspaces and
algebraic and geometric multiplicity.

**Lecture on Wednesday, Nov. 29**

We continued our study of eigenvalues by considering a number of
particular cases: the eigenvalues of A^{k} and A^{-1},
matrices wtih zero eigenvalues and diagonalizible matrices.

**Lecture on Friday, Dec. 1**

Tutorial sheet 9 was handed out. It is on the web.

We finished our study of eigenvalues and eigenvectors by considering the properties of symmetric matrices and orthogonal diagonalization (section 7.3 of the text).

This is the last lecture of the course. Term test 3 is on Monday.

**Monday, Dec. 4**

Term test 3 was held today. The marked tests should be available eraly next week. Check this website to make sure when they are ready.