Syllabus for Linear Algebra II

Professor: Mark McClure

Course purpose

With a prerequisite of Linear I, the expectation for this course is that you

• are familiar with linear systems of equations and can express those systems using the algebra of matrices,
• can use Gaussian elimination on matrices to solve linear systems of equations,
• understand the basics abstract vector spaces and linear transformations
• have heard that the finite dimensional vector spaces are exactly the Euclidean spaces \(\mathbb R^n\) and that a linear transformation mapping \(R^n \to R^m\) can be expressed as \(\vec{x} \to A\vec{x}\) where \(A\in \mathbb R^{m\times n}\),
• have heard that there are these things called eigenvalues and eigenvectors satisfying \(A\vec{x} = \lambda \vec{x}\), and
• have some experience reading and writing proofs.

Not surprisingly, we will further develop all of these things in this second semester class. We will not, however, simply jump in at the end of Linear I. Rather, we will start over with a major focus on the applied perspective. Thus, in this class, we will:

• meet several interseting, real world problems,
• need to deal with large sytems - potentially, thousands of varibles,
• use the computer computer to solve problems, and
• study the algorithms that drive the computer.

Thus, we will not use the computer as a Black Box; understanding the algorithms will require . When we do so, it turns out that the very foundations of linear algebra need to be reconsidered. We will need to re-examine Gaussian elimination, for example, to minimize the amount of numerical error that is introduced. The concepts of determinant and inverse, while still important conceptually, turn out to be essentially useless from the numerical perspective.

Materials

• Text: We will use Applied Linear Algebra by Olver and Shakiban. This is a reasonably priced text from Springer's outstanding Undergraduate Texts in Mathematics series that focus on exactly our subject matter.
• Technology:
  • Calculators: We won't be using calculators and they will not be permitted on quizzes or exams.
  • Python: We'll write computer code in Python. I recommend you get Anaconda, a free Python distribution with loads of scientific libraries pre-installed.
  • LaTeX: I'll expect you to type up a few HW assignments using LaTeX. I recommend:
    • MacTeX for Macs
    • MikTeX for Windows
    • TeXLive for Linux
    • Overleaf online
  • Our online forum : I've set up an online forum Linear Talk where we can discuss all kinds of aspects of linear algebra.

Evaluation

The standard 90-80-70-60 scale will guarantee you an A, B, C, or D. However, it is quite likely that the final scale will be shifted down from this. You will be apprised of your standing as the term progresses.

• Exams: There will be two exams during the semester worth about 100 points apiece. Likely dates for the exams are:
  • Wednesday, October 2 or Thursday, October 3
  • Wednesday, November 13 or Thursday, November 14
• Quizzes: There will be two quizzes - each, three weeks ahead of an exam:
  • Wednesday, September 11 or Thursday, September 12
  • Wednesday, October 23 or Thursday, October 24
• Knowledge Assessments: All students will take knowledge assessments motivated by the grant. They will be required of everyone in the course but will be essentially stress-free. You will earn 20 points for doing each of these. The dates of these are:
  • Tuesday, September 3 or Wednesday, September 4 and
  • Tuesday, November 20 or November, 21
• Final exam: There will be a comprehensive, final exam worth around 180 points at 11:30 AM on Monday, December 10.
• Homework: There will be several types of homework:
  • Daily textbook assignments: I'll typically assign problems for you to think about from each section that we cover. These provide important practice but will not be collected. Hopefully, some of these will be discussed on Linear Talk.
  • Online HW: We'll use the freely available MyOpenMath system for online HW, when the problems are purely computational.
  • Typed up HW: Several times during the term (perhaps 3 to 5 times), I'll ask you to type up your solution to a textbook problem for 10 to 20 points.
  • Forum assignments: Occasionally, I'll post questions in the Assignments Category of our forum that will count for 10 to 20 points. Often, these questions will involve computational work on the computer.
• Late work: In general, I don't accept late work.
• Cheating: I don't deal with cheating. If I suspect cheating strongly enough, I simply refer you to the provost and fail you for the class.

Advice

• Learning Mathematics: I believe that mathematics is a wondrous but challenging field. I assume that most people in this class have interest in mathematics and appreciate its unique challenges. There will be times of frustration ahead. Buckle down and work hard.
• Help: You are not in this endeavor alone. You have four major sources of help:
  • Me: I like to talk to people about mathematics. That's why I chose this profession. My full schedule with office hours is shown below. You will almost always find me in my office during my office hours but please feel free to approach me any time you have questions.
  • Your classmates: Most people learn mathematics best by talking through it with others. You will find that you can both learn from and help your fellow classmates. You should get to know one another very well.
  • Linear Talk: A kinda combination of the previous two that never sleeps!
  • The text: It is very important that you read the text and reread the text and raise questions about the text with me and others until you understand it. We are aiming for a deeper level of understanding than in a lower level course and we are trying to learn to communicate that understanding. You should emulate the text in your mathematical writing.

Your rights and responsibilities

It's worth understanding your rights and responsibilities as a student at UNCA. One of my responsibilities is to make sure you have the information that you need to do that. Since this is common to all classes, I've got that information on this legalese document.