General Information

As announced in the course syllabus, the first exam of the semester will be given in class on Friday, February 16. 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 the material we have discussed from the start of the semester through class on February 9, although I will not as questions specifically about the Leontief models from section 2.6. This includes the material from sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8, 1.9, 2.2, 2.3. You are not responsible for the material from sections we have not discussed in class (1.6, 1.10, 2.4, 2.5). Here is a specific list:

1. Systems of linear equations
2. Row reduction and echelon form, pivot positions
3. Linear combinations and spans
4. Solution sets of systems of linear equations in parametric form
5. Homogeneous and inhomogeneous systems, the connection between the solutions of Ax = b and Ax = 0
6. Linear independence and dependence for sets of vectors
7. Linear transformations T: Rⁿ -> R^m and their matrices
8. Matrix operations and matrix inverses; different characterizations of invertible matrices.

1. Write a system of linear equations in matrix Ax = b form, or vector form.
2. Reduce a coefficient or augmented matrix to echelon form.
3. Parametrize the set of solutions of a system of linear equations.
4. Determine whether a vector is in the span of a given set of vectors.
5. Determine whether a given set of vectors is linearly independent or linearly dependent.
6. Determine whether the linear mapping defined by a given matrix is one-to-one or onto.
7. Determine whether a given square matrix is invertible, and if so find the inverse.

Proofs to Know

1. A set S is linearly dependent if and only if there exists some x in S such that x is a linear combination of the vectors in S - {x}.
2. Every homogeneous system Ax = 0 of linear equations with m < n (i.e. more variables than equations) has a nontrivial solution.

Review Problems

There are many good review problems in the sections we have covered. You should try a selection of the odd-numbered problems we have not done on the problem sets to practice. Note: Problems marked [M] are harder calculations (i.e. not suitable for an exam problem). The True-False questions in each section are also very good practice.

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.