General Information

As announced in the course syllabus, the third exam of the semester will be given in class next Friday, April 23. You will have the full class period to work on the exam. The format will be similar to that of the first two exams -- four or five problems, each possibly containing several parts. One question again this time may consist of several ``true - false'' questions where you must either give a short proof (if you think the statement is true), or a counterexample (if you think the statement is false).

Topics to be Covered

This exam will cover all the material since the second exam, through and including the material on the Diagonalization Theorem (Chapters 13, 14 in the text -- some of the review problems below also deal with special classes of linear mappings introduced in Chapter 12 of the text). Of course, all of this depends heavily on the material on vector spaces, subspaces, bases, dimension, matrices of linear mappings, etc. from the earlier sections of the course. You will need to have that material "under control" for this exam too. Specifically, the new topics for this exam are:

1. Gaussian and Gauss-Jordan elimination for solving systems of linear equations and inverting matrices
2. The matrix interpretation of Gaussian and Gauss-Jordan elimination (via elementary matrices E_ij(c), E_i(c), P_ij)
3. Eigenvalues and Eigenvectors
4. Determinants and the Characteristic Polynomial
5. Diagonalizability; the characterization of diagonalizable linear mappings T : V -> V.

Proofs to Know

1. If det(A) <> 0, then A is invertible (using the cofactor matrix A^cof)
2. If A_i, i = 1, ..., n are eigenvectors of T with respect to distinct eigenvalues e_i, then {A₁, ... , A_n} is linearly independent.

Review Session

Scheduling a review session this time is going to be more difficult. I have a commitment on campus (the HC Chamber Orchestra concert) on Wednesday evening (warm-up at 6, concert at 8), and another commitment off-campus on Thursday evening. I think the best solution will be to make Wednesday's class the review session.

Review Problems

Additional Problem:

A linear mapping P : V -> V is said to be a projection if P² = P.

• a) Show that P(x,y,z) = (x,0,z) defines a projection P : R³ -> R³
• b) Show in general that the eigenvalues of a projection are either e = 0 or e = 1.
• c) Show that every projection is diagonalizable, and describe the diagonal matrices you get. (Hint: think about the proof of the Dimension Theorem!)

From Smith: Chapter 12/12-16; Chapter 13/ 2, 6 c,d, 11, 12; Chapter 14/6, 7 (a nilpotent linear mapping is one that satisfies T^k = 0 for some k >= 1), 8,9,14,19,25,26,27,29