MATH 2318 – Linear Algebra I

Spring 2006 Semester

(Required)

Catalog Data 2006-2008:        

MATH 2318: 

A first course in linear algebra, including vector and matrix arithmetic, solutions of linear systems and the Eigenvalue-Eigenvector problem, elementary vector spaces, and linear transformation theory.  

Prerequisite:  MATH 2413 (MATH 2376) or current enrollment in MATH 2413 (MATH 2376).

Textbook:                         Elementary Linear Algebra with Applications, 3^rd edition, by Richard Hill          

Coordinator:                      Dr. Jennifer Daniel       

Course Objective:        

   1.      To solve systems of linear equations by using elementary equation operations or                        

            by looking at the associated matrix equation and using elementary row operations,

   2.      To find the sum, difference, or product of two appropriate size matrices.

   3.      To find the inverse of an invertible square matrix.

   4.      To find the LU decomposition of a matrix and use this decomposition to solve a linear system.

   5.      To define vector space and to use this definition to determine whether a set with two operations is                      in fact a vector space.

   6.      To ascertain whether a subset of a vector space is a subspace.

   7.      To determine if a set of vectors is a linearly independent set, a spanning set, and/or a basis for a                 vector space.

   8.      To find the column space, row space, or null space of a matrix.

   9.      To determine the coordinates of a vector with respect to an ordered basis and to find a change of                          basis matrix from one basis to another.

  10.     To ascertain whether a map between vector spaces is a linear transformation and find the matrix                           associated with a linear transformation.

  11.     To use the least squares method to find the closest solution to an inconsistent system,

  12.     To use the Gram-Schmidt process to find an orthogonal or orthonormal basis for a vector space.

  13.     To determine the determinant of a square matrix and work with the properties of determinants.

  14.     To find eigenvalues and eigenvectors of a matrix.

Prerequisites by Topic:             

Topics:

 1.        Introduction to Linear Systems and Matrices

            Gaussian Elimination     

 2.        The Algebra of Matrices

            Project on Matrix Multiplication      

 3.        Inverses and Elementary Matrices

 4.        Gaussian Eliminattion as a Matrix factorization

 5.        Transposes, Symmetry, and Band Matrices

 6.        I11 Conditioned Systems

            Project on Ill Conditioned Systems

 7.        Vectors in 2 and 3 Space 

 8.        Euclidean n Space

 9.        Subspaces, Span, Null Spaces   

10.       Linear Independence

11.       Basis and Dimension

12.       The Fundamental Subspaces of a Matrix; Rank

13.       Coordinates and Change of Basis

14.       Matrices and Linear Transformations

15.       Relationships Involving Inner

16.       Least Spares and Orthogonal Projections

17.       Orthogonal Bases and Gram-Schmidt

18.       Orthogonal Matrices, QR Decompositions,

               and Least Squares (Revisited)

            Project on Least Squares

19.       A Brief Introduction to Determinants

20.       Eigenvalues and Eigenvectors

21.       Diagonalization

22.       Symmetric Matrices

23.       Application Difference Equations

24.       Quadratic Forms

25.       Solving the Eigenvalue Problem Numerically

            Project on Direct Iteration

Schedule:      Three 50-minute or 75-minute lectures per week.  Three 50-minute exams and a 2-hour final.