Math 20 - Introduction to Linear Algebra and Multivariable Calculus
Resources

Topic-Specific Tutorials

• Lagrange Multipliers: I've posted lecture notes from the lesson on Lagrange multipliers as well as a Mathematica notebook for that lesson.

• Tangent Planes: Samik introduced you to tangent planes in his third lecture, but there is no material in our calculus textbook on the topic. Saint Louis University hosts a tangent plane Java applet useful for exploring tangent planes geometrically. Here's a tangent plane tutorial from the University of Oklahoma with some good pictures explaining how the tangent plane is determined. Unfortunately, this tutorial doesn't fully explain the algebra that gives the tangent plane equation.

• Chapter 5 Lecture Notes: I think that Chapter 5 is a little confusing (and Spring Break hit right during the middle of it), so I've written some lecture notes on the material to help it make sense. These notes cover up through the class on Monday, April 4th.

• Eigenvalues: This Java applet lets you enter a 2x2 matrix and visually determine its eigenvalues and eigenvectors by dragging an arrow around the screen. Very cool. This applet was developed for Gilbert Strang's linear algebra course at MIT. You should probably start with this demo of the applet, narrated by Professor Strang.

Mathematica

Mathematica is a computer algebra system produced by Wolfram Research. Think of it as the most versatile graphing calculator you'll ever see. It will calculate, graph, find derivatives, and much more. It can also do some pretty fast operations on very large matrices, so it might come in handy on your application project, especially if you use actual data. It's also great at plotting functions of two variables, which will come in handy as we work through the multivariable calculus material.

The good news is that Mathematica is available to Harvard students for free. One option is to download the program to your personal computer. Consult the FAS Software Download page for instructions on how to do so. Note that getting that password from the Harvard Mathematica site license people can take a day or two, so plan accordingly.

Mathematica is also available in most, if not all, of the FAS computer labs. Information on printing in the Science Center computer labs is available here.

Here's a good general introduction to Mathematica. That link leads to what's called a Mathematica "notebook." That's the file format for Mathematica files. Right-click on the link and save the target to your computer. Then open the file in Mathematica once you have Mathematica installed and running. The notebook is an interactive introduction to the basics of working with Mathematica.

Note that you can save a Mathematica notebook for later use if you are not done with it in one sitting. Just remember that when you reopen a Mathematica notebook, you will have to re-execute all of the commands in it.

For help with using Mathematica to do linear algebra, consult sections 1.8.3 and 3.7 of the Mathematica Book, available as part of the online help system in Mathematica. The Mathematica tutorial we used in class on April 11th is from Elementary Linear Algebra, third edition, by Andrilli and Hecker. If you didn't get a copy of this handout, come by my office and I will give you one. For plotting functions of two variables, look up "Plot3D" in the Mathematica help system.

Below are some of the Mathematica notebooks I have used in class. You might find these notebooks useful in your application projects. To open these Mathematica notebooks, right-click the links, select "Save As...", save the notebook to your computer, start Mathematica, then open the notebook from within Mathematica.

• §4.9 Markov Chains (the mouse problem)
• §6.6 Linear Regression (SAT and GPA data file)
• Linear Programming (the two- and three-widget problems)

Study Resources

• Gilbert Strang's Linear Algebra Lectures: MIT professor Gilbert Strang has written one of the classic linear algebra textbooks, Introduction to Linear Algebra, 3rd edition, and he has been teaching linear algebra for years, if not decades. You can see watch his linear algebra lectures from the fall of 1999 online thanks to MIT's OpenCourseWare project.

• Our Textbook's Web Site: Our textbook's web site has several resources you might find useful. It features a study guide for chapter 1, data you can download for over 800 numerical exercises from the text, case studies and applications which expand on the applications presented in the text, review sheets, and practice exams.

• Reading Your Textbook: As part of a Math Warm-Up Series talk on using math help resources effectively, I prepared this handout on tips for reading a math textbook. You might find it useful as you read your textbook before class each day.

• Office Hours: In my Math Warm-Up Series talk, I also discussed how to get the most out of your instructor's office hours. Here's a summary of that discussion. There are a lot of good reasons to visit your instructor's office hour, and you should make a point to stop by at least every other week.

• Supplemental Textbooks: There are two shelves of precalculus and calculus textbooks on reserve in the Cabot Science Library. Look for them near the reference desk on the first floor. You might find them useful as precalculus review or supplemental reading. Here's a list of some of the textbooks on these shelves.

• Math Question Center: The Math Question Center (MQC) is open for business. If you have a question about your homework or just want to find some fellow students with whom to study, the MQC is a great place to go. The MQC is staffed by the calculus CAs who are there to help you with the course material. See the MQC Web Site for more information.