Mathematics - MATH 230

Course Description

Topics covered in this course include systems of linear equations, matrices, determinants, Euclidean vector spaces, general vector spaces, inner product spaces, eigenvalues and eigenvectors, diagonalization, linear transformations and applications.

Prerequisites

(A requirement that must be completed before taking this course.)

Course Competencies

Upon successful completion of the course, the student should be able to:

• Calculate an expression involving matrix operations including addition, subtraction, multiplication, scalar-multiplication, and transposition.
• Calculate the determinant of a given matrix using various techniques including cofactor expansion, row reduction, and shortcuts for small or triangular matrices.
• Calculate the multiplicative inverse of a given square matrix using various techniques including Gauss-Jordan Elimination and the adjoint-determinant method.
• Determine the solution of a system of linear equations using various techniques including Gauss, Gauss-Jordan, matrix-inverse, and Cramer methods.
• Calculate an expression involving vector operations, including addition, subtraction, scalar-multiplication, dot-multiplication, cross-multiplication, magnitudes, and parallel and perpendicular projections.
• Figure out equations for a line or plane in three-dimensional Euclidean space, given certain facts about the line or plane.
• Determine whether a given set of objects and operations constitute a vector space or subspace by using the relevant axioms.
• Determine whether a given set of vectors is linearly independent, whether it spans a given vector space, and whether it constitutes a basis for the vector space.
• Construct a basis for the null space of a given system of linear equations.
• Construct a basis for the linear span of a given set of vectors.
• Calculate the coordinates of a given vector relative to a given basis.
• Calculate the change-of-basis matrix for transitions from one given basis to another.
• Translate the coordinates of a given vector from one basis to another by using the change-of-basis matrix.
• Determine whether a given scalar function on a given vector space constitutes an inner product by using the relevant axioms.
• Calculate lengths, distances, and angles between vectors using a specified inner product.
• Construct an orthonormal basis for a given set of vectors in an inner product space by using the Gram-Schmidt method.
• Determine whether a given function between vector spaces constitutes a linear transformation by using the relevant axioms.
• Construct a basis for the kernel or for the range of a given linear transformation.
• Calculate the matrix that represents a given linear transformation relative to a given pair of bases.
• Calculate the values of a given linear transformation (or of its inverse) by using the matrix that represents it.
• Determine the eigenvalues and eigenvectors of a given linear transformation or matrix.
• Determine a diagonalized or orthogonally diagonalized form for a given linear transformation or matrix.
• Calculate powers of a given square matrix by using a diagonalized or orthogonally diagonalized form.
• Apply matrix methods to solve selected types of practical problems involving networks, curve-fitting, directed graphs, Markov chains, linear differential equations, conic sections, or quadric surfaces.

Course Schedule