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Credit hours

4 credits

Prerequisites

Math 2141 with at least a grade of C

Course Description

This course studies linear systems, matrices, determinants, finite dimensional vector spaces, linear transformations, eigenvectors and eigenvalues, and applications.

Course Objectives

1. Apply fundamental matrix operations to solve structured mathematical problems.
2. Formulate and solve real-world problems in fields such as physics, computer science, and economics using linear algebraic models.
3. Interpret matrix transformations and systems of equations geometrically as mappings and intersections in two and three dimensions.

Learning Outcomes

1. Solve systems of linear equations by using row operations, including general solutions, and interpret their results
2. Represent linear systems in different formats
3. Describe properties of linear systems using vectors, including basis vectors, linear independence of vectors, and linear combinations of vectors
4. Perform matrix and vector operations
5. Determine when a matrix is invertible and calculate its inverse
6. Compute and interpret determinants of matrices
7. Compute rank and null space of a matrix
8. Demonstrate understanding of and use linear transformations
9. Demonstrate an understanding of vector spaces and subspaces
10. Compute eigenvalues and eigenvectors

Course Topics

I. Linear Equations in Linear Algebra

1. Systems of Linear Equations
2. Row Reduction and Echelon Forms
3. Vector Equations
4. The Matrix Equation Ax=b
5. Solution Sets of Linear Systems
6. Applications of Linear Systems
7. Linear Independence
8. Introduction to a Linear Transformation
9. The Matrix of a Linear Transformation

II. Matrix Algebra

1. Matrix Operations
2. The Inverse of a Matrix
3. Characterizations of Invertible Matrices 

III. Determinants

1. Introduction to Determinants
2. Properties of Determinants
3. Cramer’s Rule, Volume, and Linear Transformations 

IV. Vector Spaces

1. Vector Spaces and Subspaces
2. Null Spaces, Column Spaces, Row Spaces, and Linear Transformations
3. Linearly Independent Sets; Bases
4. Coordinate Systems
5. The Dimension of a Vector Space
6. Change of Basis

V. Eigenvalues and Eigenvectors

1. Eigenvalues and Eigenvectors
2. The Characteristic Equation

VI. Orthogonality

1. Inner Product, Length, and Orthogonality

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