General Information

As announced in the course syllabus, the third exam of the semester will be given in class next Friday, April 30. 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, starting with section 2.6 (isomorphisms and inverse matrices) through and including the material on the dot product from Discussion 4. 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. Isomorphisms (invertible linear mappings), matrix inverse, and the Gauss-Jordan procedure inverting matrices (the reduction [A|I] -> [I|A^-1] (section 2.6)
2. Change of basis, similar matrices (section 2.7)
3. The determinant and its properties (Chapter 3)
4. Eigenvalues and eigenvectors, the characteristic polynomial, etc. (section 4.1)
5. Diagonalizability; the characterization of diagonalizable linear mappings T : V -> V (section 4.2)
6. The dot product in Rⁿ and its properties (section 4.3 and Discussion 4).

Proofs to Know

1. If V,W are finite-dimensional vector spaces, then V,W are isomorphic if and only if dim(V) = dim(W).
2. If A is an n x n upper triangular matrix, then det(A) = a₁₁a₂₂ ... a_nn (the product of the diagonal entries) -- proof by mathematical induction.
3. If x_i, i = 1, ..., n are eigenvectors of T with respect to distinct eigenvalues lambda_i, then {x₁, ... , x_n} is linearly independent.

Review Session

If there is interest, we can have an evening review session next week. Wednesday (after the Pi Mu Epsilon induction and majors' dinner), or Thursday are possible this time.

Suggested Review Problems

• Go back over your corrected papers for Problem Sets 6, 7, 8 and Discussion 4, and solutions handed out in class. Be sure you can do any problems that gave you trouble the first time.
• From Chapter 2 Supplementary Problems: 1ghi, 2,3 (apply (2.7.5), 6,7,9 (apply (2.7.5))
• From Chapter 3 Supplementary Problems: 1bcdeghij, 3,4,9
• From Chapter 4, section 3: 4, 6, 7, 9
• From Chapter 4 Supplementary Problems: 1abdef,2,3,4,6abc