The Final Exam will be in the usual classroom on Wednesday, May 13, 2009 from 8:00 -10:15 a. m. Note that you will have an extra 15 minutes to work, so there should be ample time to complete and check all the problems you know how to do. Although the exam is cumulative, it will emphasize the material covered since Exam III (that is, eigenvalues, eigenvectors, and diagonalization from chapter 7), and will focus on the topics indicated in the description below. There are 75 points possible.

Some problems may repeat or be very similar to problems on the three in-class exams, so it might be a good idea to study the solutions to those. As usual, some exam problems may be very similar to homework problems, while others will draw upon the material presented in the lectures.

Calculators or other mechanical assistance are not needed and are not to be used. Blank paper will be provided, so all you will need is something to write with. Please write your solutions on the blank paper (you may have as many sheets as you need, and may put the problems in any order). Please hand in your exam paper along with your solutions.

On the exam day, please sit with an even number of seats to the right of you in your row (that is, in alternating columns), so as not to be distracted by nearby students.

The following topics are very likely to appear, although the exam is not limited to these topics:

1. Singular and nonsingular matrices, finding inverses.
2. Rank and nullity, what they are and what they tell us about solutions of a linear system AX = B.
3. Inner products and orthogonality (fairly minimal coverage of this topic).
4. Linear independence and span.
5. Linear transformations. Definition, basic examples, kernel and range.
6. Determinants, calculation using row operations and using cofactors.
7. Calculation of eigenvalues, eigenvectors, and eigenspaces, diagonalization of matrices.
8. Characteristic polynomial of a matrix, definition and calculation.

The following topics do not appear, at least not explicitly: verification of basic properties of matrix multiplication, the formal definition of a vector space, infinite-dimensional vector spaces, inner products on vector spaces of functions, Theorem 5.2 (the description of an inner product using a symmetric matrix C), the Cauchy-Schwarz inequality, one-to-one functions, onto functions, permutations and inversions, the formal definition of the determinant, adjoint matrix and the formula for the inverse of a matrix in terms of the cofactors.