General Information

As announced in the course syllabus, the final midterm exam of the semester will be given in class on Friday, April 27. 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. Again, one question this time may (that is, will) 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 the material we have discussed starting with the week of Exam 2, through and including class on April 20. This is the material from sections 4.4, 4.5, 4.6, 4.7, 5.1, 5.2, 5.3, 5.4, 6.2, 6.7 in the text. You are not responsible for the material from sections of the text we have not discussed in class. Here is a specific list of the new topics.

Important Note: As you know very well by now, linear algebra is a very ``cumulative'' subject. You will need to be prepared for questions that, while dealing with the new topics we have studied since Exam 2, also make use of concepts and techniques we introduced earlier. In other words, you may also want to consult the Review Sheets for Exams 1 and 2 and review your class notes from earlier in the semester to solidify ideas about echelon form, properties of solutions of systems of linear equations, linear combinations and spans, linear independence, linear transformations, matrix inverses, vector spaces, subspaces, linear mappings, bases and dimension, determinants and so forth (as needed).

1. The column and null spaces of an m x n matrix.
2. Bases -- Know how to find bases for the column and null spaces of a given matrix in particular.
3. Coordinate systems in a vector space V, the coordinate vector of a vector with respect to a given basis.
4. Change of basis matrices.
5. The matrix of T : V -> W with respect to bases B for V and C for W.
6. The change of basis formula for [T].
7. Eigenvalues and eigenvectors.
8. The characteristic equation.
9. Diagonalization and diagonalizability, similarity of matrices.
10. Dot products, orthogonality, orthogonal projections.

``Big'' Proofs to Know

1. If λ₁, ... , λ_k are distinct eigenvalues of a matrix A and v₁, ... , v_k are corresponding eigenvectors, then {v₁, ... , v_k} is a linearly independent set.
2. If A,B are similar matrices, then they have the same eigenvalues with the same multiplicities.

Review Problems

Section 4.4: 3, 7, 9, 11, 13, 19, 21, 23, 25

Section 4.5: 19, 20, 21, 29, 30

Section 4.7: 1, 5, 9, 13, 19

Section 5.1: 5, 9, 15, 19, 21, 22, 23, 27, 29, 31, 33,

Section 5.2: 5, 7, 13, 17, 18, 19, 21, 22

Section 5.3: 5, 13 (be prepared, though -- I may not give you the eigenvalues on the exam), 21, 22, 23, 24, 25, 26, 27

Section 5.4: 1, 3, 5 (also be prepared to do problems like this using different bases in the domain and target spaces), 9 (ditto), 13, 15, 17, 19-24 (note: these require the ideas of sections 5.1 - 5.3),

Section 6.2: 5, 9 (use dot products to compute the coefficients in the linear combination), 23, 24.

Sample exam questions, some of the types of problems I might ask, is posted on the course homepage. (Note: this quite a bit longer than the actual exam will be.)

Disclaimer: Of course the actual exam problems may be somewhat different, I may ask things in different ways, combine topics in a single question, and so forth.

Review Session

If there is interest, I would be happy to run an evening review session on Wednesday evening next week (I have other commitments Tuesday and Thursday).