MATH 2010 - Introduction to Linear Algebra

3 sem hrs cr

This course is a study of matrices, systems of linear equations, determinants, vectors, vector spaces, eigenvalues, eigenvectors, and other selected topics. Prerequisite: MATH 1910  

In rare and unusual circumstances, a course prerequisite can be overridden with the permission of the Department Lead for the discipline.

Formerly/Same As (Formerly MAT 2830)

Transfer (UT) or Non-Transfer Course (UN): UT

By the end of the course, students will be able to…

• teach the skills needed to solve systems of equations using various matrix methods.
• familiarize students with theoretical aspects of matrix operations, including proofs.
• teach skills needed to evaluate and use determinants.
• teach vector skills that are necessary in other academic courses.
• introduce the student to abstract mathematical thinking using the concept of a vector space.
• teach the student how to find and change bases of vector spaces.
• teach the student how to find eigenvectors and eigenvectors of a matrix.
• make the student familiar with certain applications of matrix theory.

Course Objectives

Throughout the course, students will have the opportunity to…

• understand basic terminology and concepts regarding solutions of systems of linear equations.
• solve systems of linear equations using Gaussian elimination and Gauss-Jordan elimination.
• perform matrix operations, including addition, subtraction, multiplication, and transpose.
• use matrix operations to find the inverse of a matrix.
• understand algebraic properties of matrices.
• use elementary row operations to find the inverse of a matrix.
• perform operations with diagonal, triangular, and symmetric matrices.
• evaluate determinants of matrices by cofactor expansion.
• evaluate determinants of matrices by row reduction.
• use algebraic properties of determinants to solve problems.
• solve systems of linear equations using Cramer’s Rule.
• understand basic terminology and geometric and algebraic operations on vectors in 2, 3, and n dimensions.
• find the norm of a vector and perform vector arithmetic.
• find dot products of two vectors and the angle between two vectors and understand the geometric interpretation of the dot product.
• understand the idea of orthogonality and solve problems involving perpendicular vectors, including projections and distances.
• find the cross product of two vectors and apply to geometric problems.
• understand and verify the ten axioms of a vector space.
• recognize and verify when one vector space is a subspace of another.
• understand the concepts of linear independence and dependence of sets of vectors and spanning sets.
• find bases for vector spaces.
• find the dimension of a vector space and understand its algebraic and geometrical significance.
• change the basis of a vector space.
• find the row, column, and null vector spaces of a given matrix and understand their relationships to systems of linear equations. (optional)
• determine the rank of a matrix and understand its implications. (optional)
• find the eigenvalues and eigenvectors of certain matrices.
• understand Theorem 5.1.6, which ties together most of the main ideas of the study of the subject of linear algebra.
• define and be able to identify inner product spaces. (optional)
• construct an orthonormal basis for a vector space using the Gram-Schmidt process.
• solve systems of linear equations.
• recognize and apply orthogonal square matrices. (optional)
• understand general linear transformations and be able to perform them. (optional)