Introductory Linear Algebra
Spring 2014

This is the official webpage for Introductory Linear Algebra 640:250, Section C2. Check back here for homework assignments, announcements, and other course materials.

Comments about the lecture

1/23: Netpbm is a class of file formats for storing images as matrices, like we mentioned in class today. Check out the wikipedia page, and this description of the pgm format. If you have a picture in this format, what can you do with it? Could you write a program to take the transpose of a pgm image?

1/27: In your homework, you will be applying row operations to matrices, and looking at matrices which may or may not be in row reduced echelon form (r.r.e.f.). Check out this online application which gives you the steps for turning a matrix into r.r.e.f. Try some small examples to see what r.r.e.f. looks like. Can you think of a methodical procedure to turn any 2x3 matrix into r.r.e.f. with elementary row operations?

1/30: In lecture, we discussed a procedure which turns any matrix into a matrix in row reduced echeclon form called Gaussian elimination. It is interesting to note that, even in the development of European mathematics, Carl Friedrich Gauss was not the first mathematician to write down a general procedure for row reducing a matrix. You can read more about the history and development of solving systems of linear equations with elimination methods in this article by Joseph Grcar.

2/3: Due to inclement weather, class was cancelled. Please have your homework ready to hand in on Thursday. Please ensure that all pages of your assignment are stapled together, or similarly securely attached together. We will resume on Thursday 2/6 with section 1.6.

2/6: The next homework assignment is due on Thursday 2/13, and will consist of the problems from sections 1.6 and 1.7.

2/13: Due to inclement weather, class was cancelled. The homework assignment is now due on Monday 2/17. The first midterm is still scheduled for 2/27 during our usual class time. The topic of LU decomposition has been dropped. It is a very useful thing to know if you are ever doing computer calculations to solve systems of linear equations. I suggest you look it over if you have time.

2/17: Here is an old midterm for you to look at in your preparation for the first midterm. Check back for more exam preparation materials.

3/6: The mathematician Charles Lutwidge Dodgson discovered yet another method of computing determinants. You may know this mathematician by his pen name, Lewis Carrol, or from his book Alice's Adventures in Wonderland. Wikipedia has a great introduction to his method, now called Dodgson condensation. Try out some of the examples, to see if you like Dodgson's method of computing determinants.

3/13: NASA has a robotic crewmember aboard the International Space Station called Robonaut 2 (R2). In this news release, NASA says it is sending up legs for R2 on the upcoming SpaceX resupply ship. In the article, they say each of R2's legs have 7 joints with one claw to grab things, so the configuration space of R2's legs is 7+1 = 8 dimensional. In the video accompanying the article, you can see some of the maneuvers R2 is programmed to do. How would you try to program R2 to do some simple maneuvers? How can you use linear algebra to describe the configuration space of R2's legs? Can you draw a picture describing one of R2's legs?

3/27: I posted an excerpt from Timothy Gowers's book Mathematics, a Very Short Introduction in the resources section of the Sakai page for this course. The passage addresses some common misconceptions about the concept of high dimensional spaces.

3/31: The second midterm is fast approaching! Here is a good review sheet with plenty of practice problems. Another midterm can be found here. You can skip questions on linear transformations and LU decompositions.The exam will be formatted in a fashion similar to the first midterm. Please bring plenty of questions to class on Thursday, and the review sessions next week.

4/3: There are many useful online tools for practicing row reduction, finding bases, nullity, rank, etc. on the website Linear Algebra Toolkit.

4/14: Today we talked about the google PageRank method for ranking webpages. Take a look at the original source and see how our description compares.

4/15: Here is a good review sheet to look at as you prepare for the final exam.

Course information

Office hours: Thursdays 5:00PM–6:30PM (17:00–18:30)
Office: Hill 512
Textbook: Lawrence Spence et al., Elementary Linear Algebra: A Matrix Approach 2e. Upper Sadle River, New Jersey: Pearson Education, 2008.


The topics of each lecture are given in the table below. You are expected to read through the sections before class, and work through the examples in the sections. We will go over the material in class, and work more examples. Be prepared to ask questions about what you did not understand.


The suggested homework problems for each section can be found here (.pdf). You are expected to work through, and turn in all of the problems in regular weight font (non-bold). A selection of these problems will be graded on content. I will make the selected problems known to you in class. The homework sets are to be turned in every Monday, and consist of the problems from the previous week's lecture topics. No late assignments will be accepted. Instead, your two lowest homework grades will be dropped. Homework accounts for 15% of your overall grade.

After you read the section when preparing for lecture, look at the True-False questions and try to answer them. We will discuss these further in class.


Most classes will end with a short quiz on the material covered in lecture. If you arrive to class more than 30 minutes late, you will not be allowed to take the quiz. No make-up quizzes will be offered. Instead, your lowest two quiz grades will be dropped. Quizzes account for 15% of your overall grade.


This class will have two midterms and one final exam. The midterm dates are listed in the table below. These dates may change. No calculators, books, or calculation aids will be allowed during the exam. Each midterm accounts for 20% of your grade, and the final exam accounts for 30%.


• It is very important that you are actively reading the sections before each lecture. That means, you should read each section with pen and paper in hand, working out the computations for yourself.
• Work through all of the suggested homework problems. If you finish them, work on the others!
• Don't be afraid to ask questions in lecture. Try to come up with questions while you are reading the section.
• The material builds on itself. Be sure to ask questions early on, so that you are prepared for the more difficult material later.

Schedule of Lectures

Date Lecture Sections Topic, and selected problems
1/23 1 1.1, 1.2 Matrices, Vectors, and Linear Combinations 1.1: 5, 82; 1.2: 19, 31, 37
1/27 2 1.3 Systems of Linear Equations 9, 47, 53
1/30 3 1.4 Gaussian Elimination 11, 19, 81
2/6 4 1.6 Span of a Set of Vectors 17, 39, 70
2/10 5 1.7 Linear Dependence and Linear Independence 25, 33, 57
2/17 6 2.1 Homogeneous Systems, Matrix Multiplication 17, 23, 27
2/20 7 2.3 Invertibility and Elementary Matrices 13, 31, 69
2/24 8 2.4 Inverse of a Matrix
2/27 10 Midterm
3/3 11 3.1 Determinants; Cofactor Expansions 15, 23, 43
3/6 12 3.2 Properties of Determinants 17, 27, 72
3/10 13 4.1 Subspaces 9, 29, 83
3/13 14 4.2 Basis and Dimension 5, 21, 63
3/24 15 4.3 Column Space and Null Space of a Matrix 11, 65, 74
3/27 16 5.1 Eigenvalues and Eigenvectors 17, 66, 72
3/31 17 5.2, 4.4 Characteristic Polynomial 5.2: 11, 19, 85; 4.4: only 12, 14, 23
4/3 18 4.4, 5.3 Diagonalization of a Matrix 5.3: 54, 61, 65. Don't do 77–85
4/7 19 5.5 Marokov chains, google search, difference equations
4/10 20 Midterm
4/14 21 5.5, 6.1 Google search; Geometry of Vectors: 6.1: 7, 29, 95
4/17 22 6.1 Projection onto a line; Orthogonal Sets of Vectors;
4/21 23 6.2 Gram-Schmidt Process; QR factorization: 13, 29, 37
4/24 24 6.3 Orthogonal Projection; Orthogonal Complements: 5, 21, 67
4/28 25 6.4 Least Squares; Normal Equations: 1, 5, 17
5/1 26 6.5 Orthogonal Matrices: 3, 7, 9
5/5 27 Final exam review