### Review Sheet for Test 3 - Ch 1-4, 7, and Related LAMP Materials

1 8.5 x 11 sheet with writing on both sides allowed. You may put anything you want that fits on your sheet. Be sure to put the blue box on p. 221, vector space axioms 1 and 6 and their negations on your sheet. (ie the subspace axioms), and any definitions from things we have done in the past that you don't know by heart (but, you do NOT need the other vector space axioms or their negations). Calculator allowed, but not necessary. You will sometimes be asked to show by-hand steps.

Mainly study Test 1, Test 2, (review relevant problems set solution for numerous problems on the tests), WebCT quizzes 2 and 3, and Problem set 6 (see my solutions on WebCT). BE SURE TO ALSO review class notes, 7.1, 7.2, LAMP Ch 6 module 1 Section 2: Geometry of Eigenvalues up through only exercise 2.1, LAMP Ch 6 module 2 Section 1 up through exercise 1.1 only, and LAMP Ch 6 module 3 Eigenvector Analysis of Discrete Dynamical Systems, and class notes on why diagonalization makes it easy to find powers of a matrix A (from Tues the 10th class). About half of the test will be on material from Chapter 7 and related LAMP, and the other half of the test will be on Ch 1-4 and related material.

In addition to knowing how to set problems up and perform the relevant by-hand calculations (you need to be able to quickly multiply matrices by hand, quickly take determinants by hand, quickly solve systems of equations by hand and be able to find eigenvalues, eigenvectors and the matrix P, if it exists, by hand), you also need to know examples and counterexamples.

Know the one proof:
7.1 p. 391 number 36 (we did this in class) - Prove that lambda = 0 is an eigenvalue of A if and only if A is singular.

The test will be by hand, but you will be allowed to use Maple. Be sure that you know how to input, use the commands, and read and use the output from:
with(linalg);
M:=matrix...
rref(M);
eigenvectors(M);
inverse(M);
evalm(A&*B);
evalm(inverse(P)&*A&*P);