Math 2221: Linear Algebra and its Applications

Professor Mike Zabrocki

Description: In this course, we learn the basic concepts of linear algebra which has various applications. The computation aspects of linear algebra are emphasised in Math 2221 and the applications are mostly reserved for Math 2222. Throughout this course, you should be able to master performing matrix operations and apply them to solve linear systems. In addition, we cover the following topics: Linear transformations, matrix operations, determinants, vector spaces, eigenvalues and eigenvectors. Applications will be introduced as time permits.

Professor Mike Zabrocki
Office: TEL 2028
Hours: Friday 12:30-1:30 and by appointment
phone: 416-736-2100 x33980
e-mail: zabrocki at mathstat dot yorku dot ca
Class meets MWF 1:30pm in TEL 007
Text : Linear Algebra and its Applications,
written by David C. Lay
A  picture of houses from Google Sketchup
three dimensional graphics requires lots of linear algebra

Grades for the course will be based on the following criteria:

Quizzes (best 5 of 6)
Final exam 30%

I plan to cover the following sections: 1.1-1.9 (except 1.6), 2.1-2.5, 3.1-3.3, 4.1-4.7, 5.1-5.4 (as time permits)

The first quiz will be on Friday, January 11. The Midterm will be on Friday, February 22. Check the web page reguarly for announcements of subsequent quizzes and the final.

Lecture Schedule
Friday, January 4
row reduction, 2 and 3-d geometry, section 1.1, 1.2

Monday, January 7
row reduction section 1.2

Wednesday, January 9
vectors, matrix equations, section 1.3, 1.4

Friday, January 11
section 1.5 solution sets to equations
Quiz 1
Monday, January 14
section 1.7 linear independence
return Quiz 1
Wednesday, January 16
linear independence

Friday, January 18
section 1.7 linear independence
Quiz 2
Monday, January 21
section 1.8 linear transformations

Wednesday, January 23
section 1.9 matrix linear trans, 1-1 and onto
return Quiz 2
Friday, January 25
section 2.1, 1-1 and onto

Monday, January 28
section 2.2 transpose, matrix multiplication

Wednesday, January 30
section 2.3, 2.4-invertible matrices

Friday, February 1
class cancelled due to snow

Monday, February 4

quiz 3
Wednesday, February 6
section 2.4 block matricies
return quiz 3
Friday, February 8
section 2.5 LU decomposition

February 11,13,15,18
Reading week/family day
no class
Wed, February 20
section 3.1 determinants

Fri, February 22
yes, there is a ...
Mon, February 25
section 3.2,3.3 determinants, Cramer's rule

Wed, February 27
section 3.2,3.3 determinants
return midterm
Fri, February 29
section 3.2,3.3 determinants

Mon, March 3
section 4.1, vector spaces

Wed, March 5
section 4.1, vector spaces

Fri, March 7

quiz #4
Mon, March 10
section 4.2, column and null space
return quiz 4
Wed, March 12
section 4.3- linear independence, span

Fri, March 14
section 4.3, 4.4- basis

Mon, March 17
section 4.4- more bases

Wed, March 19

quiz #5
Fri, March 21
Good Friday
no class
Mon, March 24
section 4.5, 4.6 dimension and row space
return quiz #5
Wed, March 26
section 4.7 change of basis

Fri, March 28
section 5.1, 5.2 eigenvalues, eigenvectors

Mon, March 31
section 5.2, 5.3 eigenstuff and diagonalization

Wed, April 2

quiz #6
Thurs, April 3
section 5.3, 5.4 eigenstuff

Fri, April 4


(4/21/08) I am posting the unofficial final grades for the class here.

(4/16/08) I meant to get solutions for the practice posted at least a little earlier. Sorry about that. No one has even asked me for answers... Just so you are aware, many of the answers that I provide on this set of solutions are not unique. You can always choose different bases and this often changes the rest of the answers.

(4/3/08) I made up a practice for the final. The practice involves a lot of computation so might take longer than 3 hours to do from beginning to end, but it has a lot of the elements that we covered this term (but not all, just look back at some of your quizzes). You can download a copy of the practice here. I will eventually post solutions for this practice after I have given you enough of a chance to work on it. If you have been working on the practice before I have a chance to post solutions and would like me to check your answer then feel free to send me an email and I can let you know if your answer is right or not.

(3/19/08) I received an email yesterday that online course evalations for this course can be done at Please take the time to give your feedback. You should have received an email announcement about this.

(3/19/08) Quiz today see you there. It covers sections 4.1 through 4.4.

(3/7/08) I just got the final exam schedule and it seems to be set. Our exam is April 18th at 9am and will be held in VH 3003.

(3/5/08) As far as I can tell we have classes today. I will be covering more about vector spaces. There will be a quiz on Friday on Chapter 3 and the beginning of Chapter 4.

(2/29/08) We have a makeup day, Thursday April 3 because of Family Day and Friday April 4 because of Good Friday. We do not have a day scheduled for the snow day that we had.

(2/20/08) Tentatively the exam for this class has been scheduled for Friday, April 18 from 9am-12pm.

(2/20/08) The midterm covers everything including determinants. This means all sections that we covered in the book (see the table above for the complete list).

(2/1/08) 10:40am Yikes! Snow. I just checked the York website. It looks like we have a snow day. The quiz will be rescheduled for Monday.

Current Status Effective 11:30 am Friday 1 February 2008, the University has adopted Weather Emergency Procedures at the Keele and Glendon Campuses and at both the Miles S. Nadal Management Centre and Osgoode Hall Law School Downtown Centre. All day classes have been cancelled and examinations postponed. University offices and Libraries are closed. For information about possible cancellation or postponement of a scheduled special event at the University, please telephone the Faculty, department or office responsible. The suspension of normal operations at York University will remain in effect until such time as the University lifts the Weather Emergency status.

(1/23/08) I noticed that I scheduled the midterm during reading week. Don't forget that not only will we not have class during Feb 11-15 but also no class on Feb 18 due to the new and improved Ontario holiday schedule. So the midterm will now be on Februrary 22.

(1/23/08) Finished the grading of the quiz. About half of you did well, the other half not. If you need help on this stuff please come and talk with me about it BEFORE the next quiz. The next quiz will be a week from Friday.

(1/14/08) Quiz again this Friday. 2 reasons. (1) You need to be experts on row reduction as fast as possible. The quizzes were just OK. Most of the time an error in row reducing was accompanied by an error in translating between matrices and equations and we have to get rid of both types of errors. (2) Linear independence is a major topic (even if it is just one section in the book). Note the (SG) study guide note on page 70 of the book.

(1/9/08) Quiz this Friday. Do as practice the exercises from section 1.2 and 1.3.

(1/7/08) There were fewer people on the first day than I expected, but we have started already. Read the first two sections on Chapter 1 of Lay's book on Linear Algebra if you weren't there. We will have the first quiz on Friday about row reduction and solving equations.

(1/2/08) Please check frequently for messages about this course.