Information about Exam III for Math 3333-001

Mathematics 3333-001 - Linear Algebra I - Spring 2009

Information about Exam III

Exam III will be in the usual classroom on Thursday, April 30, 2009. It will cover sections 4.9, 5.3, 6.1-6.3, and 3.1-3.3. There are 52 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. Row space of a matrix, row rank, and finding a basis for the row space. Column space of a matrix, column rank, and finding a basis for the column space.
  2. Null space of a matrix. Rank and nullity, and what they tell us about the linear system AX=B.
  3. Inner products (fairly minimal coverage of this topic). Orthogonal sets and orthonormal sets.
  4. Linear transformations. Definition, examples, kernel and range.
  5. The standard matrix representation of a linear transformation from Rⁿ to R^m.
  6. The correspondence between a vector space V and Rⁿ (where n = dim(V)) obtained using an ordered basis. The matrix representation of linear transformation with respect to ordered bases of the domain and codomain.
  7. Permutations, inversions, and the definition of the determinant function. Calculating determinants using row operations and using cofactor expansion. The relation between the determinant and nonsingularity.

The following topics do not appear, at least not explicitly: inner products on vector spaces of functions, Theorem 5.2 (the description of an inner product using a symmetric matrix C), Cauchy-Schwarz inequality, one-to-one functions, onto functions, application of determinants to computing areas, the expression for the inverse of a matrix in terms of the cofactors.

1. Row space of a matrix, row rank, and finding a basis for the row space. Column space of a matrix, column rank, and finding a basis for the column space.
2. Null space of a matrix. Rank and nullity, and what they tell us about the linear system AX=B.
3. Inner products (fairly minimal coverage of this topic). Orthogonal sets and orthonormal sets.
4. Linear transformations. Definition, examples, kernel and range.
5. The standard matrix representation of a linear transformation from Rⁿ to R^m.
6. The correspondence between a vector space V and Rⁿ (where n = dim(V)) obtained using an ordered basis. The matrix representation of linear transformation with respect to ordered bases of the domain and codomain.
7. Permutations, inversions, and the definition of the determinant function. Calculating determinants using row operations and using cofactor expansion. The relation between the determinant and nonsingularity.