Description

The course provides an introduction to linear algebra and matrix theory. It is intended primarily for engineering students. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Credit will be granted for only one of the following: MATH 240, MATH 341, or MATH 461.

Prerequisites

MATH 141 and one MATH/STAT course for which MATH 141 is a prerequisite.

Topics

Systems of Linear Equations

Gaussian elimination
Echelon forms
Existence and uniqueness of solutions
Homogeneous systems

Matrices

Addition, scalar multiplication, multiplication of matrices
Elementary matrices and inversion
LU decomposition
Systems of linear equations as matrix equations
*Partitioned matrices
Determinants and their properties
Cramer's rule

Vector Spaces

Subspaces and spanning sets
Linear independence
Basis and dimension
Row and column spaces of a matrix
Rank of a matrix
Null space of a matrix

Linear Transformations

Kernel and range
Matrix representation
Change of basis and similarity of matrices

Scalar Products and Orthogonality

Cauchy-Schwarz and triangle inequalities
Length and angles
Pythagorean theorem
Orthonormal sets
Orthogonal complements of the null space and column space
Orthogonal projection
Least squares problems
Orthogonal matrices
Gram-Schmidt process and QR factorization

Eigenvalues

Complex eigenvalues
Diagonalization of matrices
Spectral theorem for symmetric (*hermitian) matrices
*Quadratic forms
Positive definite matrices
*Nonnegative matrices

Applications to include several of the following:

*Liontief Input-Output Model
*Markov Chains
*Computer Graphics
*Least squares data fitting
*Fourier Series
*Systems of Differential Equations
*Difference Equations
*Max-Min Theory for functions of several variables (Hessian Matrix)

MATLAB use

In assigned homework through the semester