General Information

As announced in the course syllabus, the second exam of the semester will be given in class on Friday, February 23. You will have the full class period to work on the exam. The format will be similar to that of the first exam -- 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 1, through and including class on February 16. This is the material from sections 2.8, 2.9, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3 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: 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 1, also make use of concepts and techniques we introduced earlier. In other words, you may also want to consult the Review Sheet for Exam 1 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, and so forth (as needed).

1. Vector subspaces of Rⁿ.
2. The column and null spaces of an m x n matrix.
3. Bases -- Know how to find bases for the column and null spaces of a given matrix in particular.
4. Dimension of a vector space, rank of a matrix.
5. The determinant of an n x n matrix, cofactor expansions.
6. Properties of determinants (in particular, the effect of the various row operations on determinants, det(A) = 0 <=> A is not invertible, det(AB) = det(A)det(B))
7. Cramer's Rule for solving Ax = b when det(A) <> 0.
8. Determinants and areas, volumes.
9. General vector spaces, subspaces, linear transformations, bases.

1. Reduce a matrix to echelon form.
2. Parametrize the set of solutions of a system of linear equations.
3. Determine whether a vector is in the span of a given set of vectors.
4. Determine whether a given set of vectors is linearly independent or linearly dependent.
5. Determine whether the linear mapping defined by a given matrix is one-to-one or onto.
6. Determine whether a given square matrix is invertible, and if so find the inverse.
7. Find bases for the null space or column space of a given matrix.
8. Compute determinants by cofactor expansion (with row operations to simplify).
9. Solve linear systems by Cramer's Rule.
10. Show a given subset of a vector space is, or is not, a vector subspace.

``Big'' Proofs to Know

1. If V is a vector space, S is a finite subset of V with Span(S) = V, and T is a linearly independent subset of V, then |T| <= |S|. Also know how to use this fact to show that if V is a vector space with some finite basis, then all bases have the same number of elements. (Recall, this is the justification for the definition of the dimension of a vector space.) See class notes -- this result is not proved in exactly this form in the text.
2. Cramer's Rule: If A is invertible, the solution of Ax = b is given by x_i = det(A_i(b))/det(A) (where A_i(b) is the matrix obtained from A by replacing column i by the vector b).

Review Problems

Section 2.8: 1, 5, 17, 21, 22, 23, 27, 29, 31, 33

Section 2.9: 5, 9, 13, 15, 17, 18, 19, 21, 23, 27

Section 3.1: 11, 13, 21, 23, 39, 40

Section 3.2: 1, 3, 9, 21, 25, 27, 28, 29, 39, 41, 43

Section 3.3: 5, 9, 19, 23, 31

Section 4.1: 3, 7, 9, 11, 19, 23, 24

Section 4.2: 25, 26, 35

Section 4.3: 21, 22, 31, 33, 35.

A sample exam, indicating the length of the exam and some of the types of problems I might ask, is posted on the course homepage.

Disclaimer: Of course the actual exam 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 next week before the exam. Either Tuesday or Wednesday evening would be possible.