Information about Exam II for Math 3333-001

Mathematics 3333-001 - Linear Algebra I - Spring 2009

Information about Exam II

Exam II will be in the usual classroom on Tuessday, March 24, 2009. It will cover sections 2.3-2.4 and 4.2-4.6. There are 53 points possible.

Some of the exam problems will be very similar to homework problems, while others will draw upon the material presented in the lectures. As on any exam, it is wise to start with the problems that you feel confident that you know how to do, before moving on to others.

Calculators or other mechanical assistance are not needed and are not to be used. Blank paper will be provided, so all you will need is something to write with. Please write your solutions on the blank paper (you may have as many sheets as you need, and may put the problems in any order). Please hand in your exam paper along with your solutions.

On the exam day, please sit with an even number of seats to the right of you in your row (that is, in alternating columns), so as not to be distracted by nearby students.

The following topics are very likely to appear, although the exam is not limited to these topics:

  1. Elementary row operations and elementary column operations. Elementary matrices and the correspondence between left and right multiplication and elementary row and column operations.
  2. Row equivalence, column equivalence, and equivalence of matrices. What it means, and how it relates to multiplication by nonsingular matrices.
  3. Inverses of nonsingular matrices. Calculation of the inverse using the elementary row operation method.
  4. Vector spaces. The defining properties (they need not be memorized, but you should be very familiar with them and understand what each one says). Verifying that V satisfies a property, or giving a counterexample to show that it does not.
  5. Subspaces, checking closure under addition and scalar multiplication to determine whether a subset is or is not a subspace.
  6. Span of a subset, linear independence of a subset.
  7. Bases and dimension. The two big pictures and what they tell us about how bases and dimension work.

The following topics do not appear, at least not explicitly: proof of the fact that left (or right) multiplication by an elementary matrix corresponds to performing an elementary row (or column) operation, rank of a matrix, equivalence relations (i. e. the reflexivity, symmetry, and transitivity properties), finding parametric equations for a line, section 4.1, differential equations (although vector spaces whose elements are functions might appear).

1. Elementary row operations and elementary column operations. Elementary matrices and the correspondence between left and right multiplication and elementary row and column operations.
2. Row equivalence, column equivalence, and equivalence of matrices. What it means, and how it relates to multiplication by nonsingular matrices.
3. Inverses of nonsingular matrices. Calculation of the inverse using the elementary row operation method.
4. Vector spaces. The defining properties (they need not be memorized, but you should be very familiar with them and understand what each one says). Verifying that V satisfies a property, or giving a counterexample to show that it does not.
5. Subspaces, checking closure under addition and scalar multiplication to determine whether a subset is or is not a subspace.
6. Span of a subset, linear independence of a subset.
7. Bases and dimension. The two big pictures and what they tell us about how bases and dimension work.