General Information

As announced in the course syllabus, the first exam of the semester will be given in class on Friday, February 20. 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 Section 1.5 on solving systems of linear equations, echelon forms, and consequences. Bases and dimension (Section 1.6) will appear on Exam 2. 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: Rⁿ, P_k(R), F(R) 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 ``shortcut test'' (1.2.8)
4. Linear combinations and the linear span of a set of vectors
5. Linear dependence and independence
6. Solving systems of linear equations, echelon form, etc.

Proofs to Know

1. The span of any set S in a vector space V is a vector subspace of V.
2. A set S is linearly dependent if and only if there exists some x in S such that x is linearly dependent on S - {x} (and if so, Span(S) = Span(S - {x})).
3. Every homogeneous system of linear equations with m < n (i.e. more variables than equations) has a nontrivial solution.

Review Problems

Supplementary Exercises for Chapter 1 (page 59): 1adefghijk, 2 (but for part a, replace the directions ``find a basis for and the dimension of'' by ``find a spanning set for''), 3,4,5,6,7,8a (but ignore parenthetical remark), 10.

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.