General Information

As announced in the course syllabus, the second exam of the semester will be given in class on Friday, March 26. 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. One question 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 all the material since the first exam, starting with section 1.6 (Bases) through and including Section 2.5 on compositions of linear mappings and products of matrices. Here is a specific list:

1. Bases for vector spaces, finite dimensionality, the dimension of a vector space; if V is finite-dimensional, then every linearly independent subset of V can be extended to a basis.
2. Linear mappings T : V -> W from one vector space to another -- know the definition, how to determine whether a given mapping is linear or not, and key examples like the ones from Discussion 2 and Problem Set 4 (2.1).
3. The matrix of a linear mapping with respect to choices of basis in the domain and kernel, and the properties of these matrices (2.2).
4. The kernel and image of a linear mapping. Know in particular how to determine the kernel and image of mappings defined by matrices, and how to use the kernel and image to determine whether a linear mapping is injective, surjective, both, or neither (2.3).
5. The Dimension Theorem (2.3), its proof and the consequences from (2.4).
6. Compositions of linear mappings and matrix multiplication (2.5).

Proofs to Know

1. The proof that if V is a vector space with a finite spanning set S, then any linearly independent subset T of V satisfies |T| <= |S| (1.6.10)
2. How to use number 1 to show that all bases for a given vector space V have the same number of elements (when V has a finite spanning set) (1.6.11)
3. The Dimension Theorem for linear mappings (2.3.17, which uses 2.3.15)

Suggested Review Problems

• Go back over your corrected papers for Problem Sets 3,4,5, and Discussions 2, 3. If there were any problems you did not get full credit for, be sure you understand why you didn't and how to get a correct solution.
• Supplementary Exercises for Chapter 1 (page 59): 1bcd, 3, 12;
• Supplementary Exercises for Chapter 2 (page 130): 1abcdefg,2,3,4,5,8,9,10,12,13,14.

Review Session

If there is interest, I would be happy to run an evening review session next week before the exam. Wednesday evening would be best.