## Review guide for the second exam

The exam will cover what we have done on linear algebra and material from sections 12.1-12.3 on Fourier series. Here, from the syllabus, is what I think the exam will ask about:

 Goals Students should know the principal definitions of linear algebra and recognize and work with these ideas. Students should know how to compute the rank of a matrix, the determinant of a matrix, and be able to invert and diagonalize simple matrices "by hand". Goals Students should be able to compute the Fourier coefficients of simple functions "by hand". They should know what orthogonality means for functions. Given a function's graph on [0,A] they should be able to graph the sum of the Fourier sine and cosine series on [-A,A]. They should have some qualitative understanding of what partial sums of Fourier series look like compared to the original function, including an idea of the Gibbs phenomenon.

The following resources will be helpful to students when preparing for the first exam. Please also consider the draft formula sheet and, if you wish, send me comments or corrections or suggested additions.

I will hold a review session on Wednesday, November 16, from 7 PM to 9 PM in Hill 525. This session is not intended as a substitute for your own efforts to study, and it should not be used for a first glance at many of the problems referenced below.

• Material from this semester
I hope to post scanned homework solutions for the textbook problems designated by the syllabus in sections 12.1 and 12.2 done by student volunteers. When this is done, a link will be installed here.
I will certainly ask you for the definitions of various linear algebra terms. An effort to organize such definitions is here.
A formula sheet for Fourier coefficients will be handed out. A draft of this formula sheet is here.
• Material from the fall 2004 version of the course
Here are solutions to the problems suggested by the syllabus in section 12.3.

Here is the second exam together with answers to that exam. The material to be tested on the exam in our course is the same as the material tested in this exam.

• Material from the spring 2004 version of the course
The second exam that semester covered only linear algebra. With that in mind, you might want to look at the following material. There are review problems and answers. See especially problem 14 and its answers. That problem asks for very brief definitions of common terms in linear algebra. I think students in this semester's class should be able to answer that sort of question, also. A version of the second exam I gave and answers to that exam are available. I believe most of the material is quite similar to what we have done this semester, and therefore can be useful to you. Most of the spring 2004 review problems were copied from past exams in this course.

The first seven review problems here cover material to be tested on this exam, and here are answers. Problems 1 and 2 and 3 on this exam could be asked here.

• Also from fall 2004: another second midterm exam
These problems are from another section, which just gave an exam at approximately the same logical time in the course. Thus there are linear algebra questions and some Fourier series problems.
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