Linear Algebra
MTH 266-001N
Course Content Summary
Annandale Campus
Northern Virginia Community College
Dr. Don Goral
Virginia Community
College Course Content Summary
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.
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 Gauss-Jordan 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
matrix-matrix addition, scalar-matrix multiplication, and
matrix-matrix 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 one-to-one 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 rank-nullity
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