General Information

As announced in the course syllabus, the first exam of the semester will be given in class on Friday, February 19. You will have the full class period to work on the exam. The format will be similar to that of the exams from Algebraic Structures last semester -- four or five problems, each possibly containing several parts. One question 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 start of the semester, through and including the dimension of a vector space and the examples from class on February 12. (This is Chapters 1 - 7 in Smith.) Here is a specific list:

1. The axioms for vector spaces -- showing that a set is or is not a vector space using the axioms
2. The key examples of vector spaces: R^k, P_k(R), Fun(S), l_infinity, and their properties
3. Subspaces of a vector space -- know how to show that a subset of a vector space is or is not a subspace using the definition (see page 35)
4. Linear combinations and the linear span of a set of vectors
5. Linear dependence and independence
6. Bases of a vector space and the dimension of a vector space

Proofs to Know

1. The linear span of any set E in a vector space V is a vector subspace of V.
2. A set E is linearly dependent if and only if there exists some A in E such that A is linearly dependent on E - {A} (and if so, L(E) = L(E - {A})).
3. If V is a vector space with some finite spanning set, then every two bases of V have the same number of elements. (You may state without proof any facts about solutions of systems of linear equations that you need here.)

Review Problems

From Smith: Chapter 2/12,13; Chapter 3/5,6,8 (the notation is defined in Example 4 on page 29); Chapter 4/7,8,16-20,31; Chapter 5/6,17,18,19 (assume that the vectors are linearly independent as in 18), Chapter 6/1 - 7, 9; Chapter 7/1, 5, 9.