Math 2255 -- Linear Algebra I

Math 2255, combined with the second semester course Math 2275, form an introduction to Linear Algebra. This course is for students in mathematics, but students in physics, computer science, and economics may also find the results useful. In this course we look at matrices and systems of linear equations; linear transformations and matrices; elementary matrix algebra; determinants; vector spaces; change of bases; real eigenvalues and eigenvectors; applications.

News (Last Updated Nov. 30, 2007)

Below is a summary of what we did in class, plus any relevant news and/or information.

Please check your

Math 2255 Pre-Exam Marks

If there are any problems, please contact me before the final exam.

Nov. 29, 2007 In our last class, I did a review for the final exam. HWA 8 was due in class today.
Nov. 28, 2007 We discussed Markov Chains in today's lecture (see Section 4.8). You should know what a probability vector is, a Markov chain, and a steady-state vector. I also handed out the review sheet for the final exam.
Nov. 26, 2007 I introduced a new vector space, namely, the vector space of discrete signals. I also talked about differene equations. This material was based upon Section 4.8.
Nov. 23, 2007 The information sheet for the final exam has been posted.
Nov. 23, 2007 Today's class was our second discussion on Section 4.7 and the change of basis. You should know how to find the change of coordinate matrix from one basis to another. Old HWA and MatLab assignments were passed back, HWA 7 was collected, and HWA 8 assigned.
Nov. 21, 2007 We finished Section 4.6 by relating the rank of a matrix to the Invertible Matrix Theorem. We also started Section 4.7 on the Change of Bases. You should know how to do examples discussed in class. MatLab 4 was due in class today.
Nov. 19, 2007 We covered Section 4.6 on the rank of a matrix. You should know the definition of rank, and the rank theorem. We also had course evaluations today.
Nov. 16, 2007 We spent some more time discussing dimension. You should know the Basis Theorem, and the fact that every linearly independent set of vectors can be extended to a basis.
Nov. 14, 2007 We finished our discussion of coordinate systems (Section 4.4). I also began our discussion of dimension (Section 4.5). You should know how we define dimension, and be able to compute the dimension of a subspace.
Nov. 12, 2007 Today we talked about Section 4.4 and coordinate systems. You should know what we mean when we talk about the B-coordinate of vector. I handed back HWA 5 and the 3rd MatLab Assignment. HWA 6 was due in class today.
Nov. 9, 2007 Class will be cancelled Friday. Please hand in your homework on Monday. A new HWA will be given out next week.
Nov. 7, 2007 We finished Section 4.3 today. You should know how to find a basis of the null space and column space of a matrix. Also, MatLab Assignment 3 was due in class.
Nov. 5, 2007 Today we went over Section 4.3. We generalized the notion of a linearly independent set. As well, I introduced the notion of a basis.
Nov. 1, 2007 We completed Section 4.2 on Null and Column Spaces. Just a reminder that MatLab Assignment 3 is due on Wednesday.
Oct. 31, 2007 I finished up Section 4.1 on Vector Spaces, and began a discussion of Section 4.2. In particular, I introduced two important subspaces -- the span of a set of vectors and the null space of a matrix.
Oct. 29, 2007 Today's class was an important discussion on vector spaces. A vector space generalizes the properties of R^n. I also handed back the midterm.
Oct. 29, 2007 I will be moving Monday's office hour to Tuesday, October 30, from 2:30-3:30.
Oct. 26, 2007 The Midterm was today.
Oct. 24, 2007 We finished Chapter 3 today with a look at Cramer's rule. You should know this rule, and how to use this rule to find the inverse of a matrix. Also, you should know the relationship between the determinant of a matrix and the area/volume of a geometeric object.
Oct. 22, 2007 We continued our look at determinants by considering Section 3.2. You should know how row operations can change the value of the determinant. As well, know that the deterimant of a matrix equals the determinant of its transpose.
Oct. 19, 2007 We looked at Section 3.1 on determinants. After today's class, you should know how to use the formula for computing determinants. There is no new HWA this week; instead, you should study for the midterm which is next week.
Oct. 17, 2007 Today was our last look at Chapter 2. We skipped Section 2.6, but covered Section 2.7 on applications to computer graphics. You should know how to use homogeneous coordinates to do translations. I also handed out the review sheet for the midterm next week.
Oct. 15, 2007 We discussed Section 2.5 on matrix factorizations. In particular, we discussed the LU-factorization of matrices. You should know how to use the LU-factorization to solve systems of linear equations, and how to find the LU-factorization. I also handed back HWA 3.
Oct. 12, 2007 We finished up Section 2.3 by discussing the Invertible Matrix Theorem. I then quickly talked about Section 2.4 on partitioned matrices. HWA 4 was assigned, and MatLab 2 was handed back.
Oct. 10, 2007 I introduced the notion of an inverse. You will need to know how to find the inverse of a matrix. I also handed back HWA 2.
Oct. 5, 2007 Class cancelled because I was at a conference.
Oct. 3, 2007 Today we started on Chapter 2. Section 2.1 is on matrix operations (e.g. matrix addition, multiplication). This may be mostly review for some students. Please note that class is cancelled on Friday; HWA 3 is due at my office by 8:00AM on Tuesday.
Oct. 1, 2007 We continued with our discussion of linear transformations (Section 1.9). In particular, we showed that every linear transformation is a matrix transformation. As well, we showed how to determine if a transformation is one-to-one or onto from the associated standard matrix.
Sept. 28, 2007 We went over linear transformations (Section 1.8). I also assigned HWA 3.
Sept. 26, 2007 We looked at Section 1.7 on linear independence. I introduced the notion of linear independence and dependence, and gave some examples and described some basic results. MatLab 1 was also due in class today.
Sept. 24, 2007 In today's class we discussed some applications of linear algebra to problems in economics and network flow. There are also applications to chemistry which you will read about in the textbook. Today's material was based upon Section 1.6.
Sept. 21, 2007 HWA 1 was due in class. We went over Section 1.5 which discussed how to express the solution set of a system of linear equations. I also introduced the notion of a homogeneous system. I gave out the new HWA (see to the right).
Sept. 19, 2007 We went over Section 1.4 by discussing how to represent systems of linear equations by using matrices.
Sept. 17, 2007 Today we discussed Section 1.3 which introduces vector notation.
Sept. 14, 2007 We met today in ATAC 3002 to go over our introduction of MatLab. You should try working out all the problems in the practice tutorial before moving onto the lab assignments. Please note that the first lab is now due Sept. 26 (one week later). HWA 1 was also given out.
Sept. 12, 2007 In today's class we finished our discussion row reduction (Section 1.2). I introduced the pivot position and pivot column. I also explained in detail the row reduction algorithm. Note For Friday, we will meet in ATAC 3002 to be introduced to MatLab. Bring your MatLab tutorials with you.
Sept. 10, 2007 I finished Section 1.1 on system of linear equations. I gave an example of an inconsistent system, and a consistent system with an infinite number of solutions. You should also know what the echelon and reduced echelon form of a matrix is.
Sept. 7, 2007 Today was the first day of class. I began by going over the course information sheet (you can find a copy to the right). I then started on Section 1.1 on Systems of Linear Equations. You should know what a system of linear equations is, and what it means for a system to be consistent or inconsistent. I also gave a quick introduction on how to solve a system of linear equations.
Aug. 8, 2007 The webpage was linked to the main webpage. Also, if you want to hunt the internet for a used copy of the textbook, the information about the book is to the right.

Course Information

Instructor: Adam Van Tuyl

Office: RB 2015
Office Hours: M 2:30-3:30 and W 3:30-4:30
Email: avantuyl AT sleet.lakeheadu.ca

Place and Time:

Class: MWF 12:30-1:30 in RB 3044

Textbook

Linear Algebra and its applications
David Lay

Homework Soutions

All the solutions to the homework and tests are posted electronically on eRES, the electronic reserves of Lakehead's library. The solutions are PDF files.

Homework Assignments

Homework is given out every Wednesday, and will be due, in class, the following Wednesday. Assignments must conform to the guidelines in the course outline. Assignments are posted below.

Assignment 1 (Due: Sept. 21)

Sec. 1.1 -- 4, 14, 22, 24, 28, 30
Sec. 1.2 -- 2, 10, 12, 16, 22, 24, 26

Assignment 2 (Due: Sept. 28)

Sec. 1.3 -- 8, 14, 16, 24, 30
Sec. 1.4 -- 8, 18, 22, 24, 32
Sec. 1.5 -- 6, 12, 16, 24, 26

Assignment 3 (Due: Oct. 5)
NOTE: Since class is cancelled on Oct. 5, will be due
at 8:00AM at my office on Tues. Oct. 9

Sec. 1.6 -- 2, 4a, 8, 12ab
Sec. 1.7 -- 6, 14, 20, 22, 36
Sec. 1.8 -- 6, 14, 20, 22abcd, 24

Assignment 4 (Due: Oct. 19)

Sec. 1.9 -- 2, 16, 20, 24, 26
Sec. 2.1 -- 2, 12, 16, 20, 28
Sec. 2.2 -- 2, 10abcd, 14, 32
Sec. 2.3 -- 8, 12, 14, 28

Assignment 5 (Due: Nov 2)

Sec. 3.1 -- 4, 10, 18, 38, 40
Sec. 3.2 -- 6, 14, 28, 36, 40
Sec. 3.3 -- 4, 10, 12, 20, 28

Assignment 6 (Due: Nov 9)

Sec. 4.1 -- 2, 6, 10, 18, 24, 26
Sec. 4.2 -- 3, 10, 16, 24, 26abcde, 30

Assignment 7 (Due: Nov 23)

Sec. 4.3 -- 4, 6, 14, 22, 24, 30
Sec. 4.4 -- 4, 6, 14, 16, 22
Sec. 4.5 -- 4, 14, 20, 22

Assignment 8 (Due: Nov 30)

Sec. 4.6 -- 4, 8, 12, 16, 18, 30
Sec. 4.7 -- 2, 4, 8, 10, 12, 14

MatLab

We will use the computer program MatLab to solve problems in Linear Algebra. For further information:

MatLab Information Sheet

A list of the due dates of tutorials are below. (The actual tutorials will not be available online.)

Practive Row Operations and Exchange Economy and Homogeneous Systems
(Due Sept. 26) New date!
Visualizing Linear Transformations of the Plane
(Due Oct. 10)
Elementary Analysis of the Spotted Owl Population
(Due Nov. 7)
The Adacency Matrix of a Graph
(Due Nov. 21)

To do these assignments, you will need the

laytoolbox_matlab data set.