# MATH 215 Linear Algebra

**Credit Hours:** 3

**Prerequisite:** MTH 136

**General Education: **Quantitative Competence

**College Learning Outcomes:** 9a, 9b, 9c, 9d, 9e – Quantitative Competence and

2b – Critical Thinking

**I. Course Description:**This course studies systems of linear equations, vector spaces, linear transformations and matrices. It includes applications and theories.

**II. Purpose of the Course:**Linear algebra is valuable in illustrating a number of mathematical thinking processes that arise not only in linear algebra, but also in many other mathematical subjects. Understanding these thinking processes greatly reduces the time and frustration involved in learning advanced mathematics as well as in solving mathematical problems in general. It is also useful in solving a variety of problems arising in physics, chemistry, statistics, business and other areas.

**III. College Learning Outcomes and Objectives:**This course is designed to help students fulfill the Quantitative Competence Learning Outcome through achievement of the following learning objectives:

**(9a)**Students can formulate specific questions from vague problems, select effective problem-solving strategies, and know which mathematical operations are appropriate in particular contexts.

**(9b)**Students can perform mental calculations and estimates with proficiency, and decide when an exact answer is needed and when an estimate is more appropriate.

**(9c)**Students can use a calculator correctly, confidently, and appropriately and/or use computer software for mathematical tasks.

**(9d)**Students can use tables, graphs, spreadsheets and statistical techniques to organize, interpret and present numerical information.

**(9e)**Students can judge the validity of quantitative results presented by others.

**(2b)**Students can demonstrate the ability to reflect on issues and/or theories systematically.

**IV. Course Objectives**

1. Students should be able to perform arithmetic operations on vectors.(LO 9a-e, 2b)

2. Students should be able to solve linear equations using matrices.(LO 9a-e, 2b)

3. Students should be able to use determinants to solve linear problems. (LO 9a-e, 2b)

4. Students should be able solve problems involving vector spaces. (LO 9a-e, 2b)

5. Students should be able to solve problems using linear transformations.(LO 9a-e, 2b)

6. Students should be able to solve problems using eigenvalues and eigenvectors.(LO 9a-e, 2b)

7. Students should be able to use numerical methods to solve problems.(LO 9a-e, 2b)

**V. Topical Outline**

**A. Euclidean vectors**

1. Vectors in Euclidean space and their applications

2. Arithmetic operations on vectors

3. Lines and planes

4. Problem solving with vectors

**B. Using matrices to solve (m x n) linear equations**

1. Problems and applications of solving linear equations

2. Solving linear equations by row operations

3. Matrices and their operations

4. Problem solving with linear equations

**C. Using Matrices to Solve (n x n) Linear Equations**

1. Applications of (n x n) linear equations

2. Using the inverse matrix to solve linear equations

3. Using determinants to solve linear equations

4. Problem solving with linear equations

**D. Vector Spaces**

1. Basic properties and subspaces of vector spaces

2. Span, linear independence and basis

3. The dimension of a vector space

4. Problem solving with vector spaces

**E. Linear Transformations**

1. The matrix of a linear transformation

2. Transformations from V to V

3. Solving equations with linear transformations

4. Solving problems with linear transformations

**F. Eigenvalues and Eigenvectors**

1. Definition of Eigenvalues and Eigenvectors

2. Solving a dynamical system

3. Diagonalization

4. Problem solving with Eigenvalues and Eigenvectors

**G. Orthogonality and Inner Product Spaces**

1. Orthogonality in Rn

2. Applications to regression models

3. Inner product spaces

**H. Numerical Methods in Linear Algebra**

1. General computational concerns

2. Matrix factorizations

3. Finding Eigenvalues and Eigenvectors

4. Iterative methods

5. Using Numerical methods in problem solving