MTH 266 Linear Algebra
Course Title: MTH 266 Linear Algebra
Course Description
Covers matrices, vector spaces, determinants, solutions of systems of linear equations, basis and dimension, eigenvalues and eigenvectors. Designed for mathematical, physical and engineering science programs. Lecture 3 hours per week. 3 credits.
General Course Purpose
The general purpose is to give the student a solid grasp of the methods and applications of linear algebra, and to prepare the student for further coursework in mathematics, engineering, computer science and the sciences.
Course Prerequisites/Corequisites
Prerequisite: Completion of MTH 263 or equivalent with a grade of B or better or MTH 264 or equivalent with a grade of C or better.
Course Objectives
Upon completing the course, the student will be able to:
Matrices and Systems of Equations

Use correct matrix terminology to describes various types and features of matrices (triangular, symmetric, row echelon form, et.al.)

Use GaussJordan elimination to transform a matrix into reduced row echelon form

Determine conditions such that a given system of equations will have no solution, exactly one solution, or infinitely many solutions

Write the solution set for a system of linear equations by interpreting the reduced row echelon form of the augmented matrix, including expressing infinitely many solutions in terms of free parameters

Write and solve a system of equations modeling real world situations such as electric circuits or traffic flow
Matrix Operations and Matrix Inverses

Perform the operations of matrixmatrix addition, scalarmatrix multiplication, and matrixmatrix multiplication on real and complex valued matrices

State and prove the algebraic properties of matrix operations

Find the transpose of a real valued matrix and the conjugate transpose of a complex valued matrix

Identify if a matrix is symmetric (real valued)

Find the inverse of a matrix, if it exists, and know conditions for invertibility.

Use inverses to solve a linear system of equations
Determinants

Compute the determinant of a square matrix using cofactor expansion

State, prove, and apply determinant properties, including determinant of a product, inverse, transpose, and diagonal matrix

Use the determinant to determine whether a matrix is singular or nonsingular

Use the determinant of a coefficient matrix to determine whether a system of equations has a unique solution
Norm, Inner Product, and Vector Spaces

Perform operations (addition, scalar multiplication, dot product) on vectors in Rn and interpret in terms of the underlying geometry

Determine whether a given set with defined operations is a vector space
Basis, Dimension, and Subspaces

Determine whether a vector is a linear combination of a given set; express a vector as a linear combination of a given set of vectors

Determine whether a set of vectors is linearly dependent or independent

Determine bases for and dimension of vector spaces/subspaces and give the dimension of the space

Prove or disprove that a given subset is a subspace of Rn

Reduce a spanning set of vectors to a basis

Extend a linearly independent set of vectors to a basis

Find a basis for the column space or row space and the rank of a matrix

Make determinations concerning independence, spanning, basis, dimension, orthogonality and orthonormality with regards to vector spaces
Linear Transformations

Use matrix transformations to perform rotations, reflections, and dilations in Rn

Verify whether a transformation is linear

Perform operations on linear transformations including sum, difference and composition

Identify whether a linear transformation is onetoone and/or onto and whether it has an inverse

Find the matrix corresponding to a given linear transformation T: Rn > Rm

Find the kernel and range of a linear transformation

State and apply the ranknullity theorem

Compute the change of basis matrix needed to express a given vector as the coordinate vector with respect to a given basis
Eigenvalues and Eigenvectors

Calculate the eigenvalues of a square matrix, including complex eigenvalues.

Calculate the eigenvectors that correspond to a given eigenvalue, including complex eigenvalues and eigenvectors.

Compute singular values

Determine if a matrix is diagonalizable

Diagonalize a matrix
Major Topics to be Included
Matrices and Systems of Equations
Matrix Operations and Matrix Inverses
Determinants
Norm, Inner Product, and Vector Spaces
Basis, Dimension, and Subspaces
Linear Transformations
Eigenvalues and Eigenvectors