# Linear Algebra

Prerequisites: Qualifying math score and completion of Calculus BC or equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit or one semester of college credit equal to or greater than an AP class

Course Code: LIN

## Course Description

Description

This course presents the main concepts and terminology of linear algebra. It is a full introductory linear algebra course equivalent to a first-year college linear algebra course. This course covers the entire syllabus from the Johns Hopkins semester-based, in-person Linear Algebra course, plus several additional topics. For the projects, students will be introduced to mathematical typesetting with LaTeX.

### Topics include:

• linear equations
• matrix algebra
• determinants
• vector spaces
• eigenvalues
• orthogonality
• least squares
• symmetric matrices

As illustrated throughout the course, the topics presented play an essential role in areas such as computer science, engineering, environmental science, economics, statistics, business management, and the social sciences. This course provides an excellent foundation for Multivariable Calculus.

Assignments are based on a textbook that is purchased separately by the student. Students are expected to read from the text regularly.  After attempting the suggested practice problems, students attempt the graded homework problems.  Questions and hints can be obtained by direct weekly communication with instructors who provide continual feedback on their work. Students are strongly encouraged to work on the course at least 1 hour a day, 5 days a week, and email their instructors at least once per week.

Image from Math Insight

## Materials Needed

A textbook purchase is required for this course.

Linear Algebra and its Applications, 4th edition, by David C. Lay.

A graphing calculator is required, such as:

• TI-83 PLUS
• TI-84 PLUS
• TI-85
• TI-86
• TI-89*

*Recommended

## List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following topics:

### Linear Equations

• Systems of Linear Equations
• Row Operations
• Echelon Form
• Existence and Uniqueness of Solutions
• Vector Equations
• The Matrix Equation Ax=b
• Balancing Chemical Equations
• Network Flow
• Linear Independence
• Linear Transformations
• Superposition Principle
• Matrix of a Transformation
• Geometric Transformations
• One-to-one Transformations

### Matrix Algebra

• Matrix Arithmetic
• Row-Column Rule for AB
• Matrix Inversion
• Inversion and Ax=b
• Properties of Inverse
• Elementary Matrices
• Row Reduction and Inverses
• Invertible Matrix Theorem
• Partitioned Matrices
• Column-Row Expansion for AB
• LU Factorization
• LU and Electrical Circuits
• Leontief Input-Output Model
• Basic Computer Graphics

### Determinants

• Definition of Determinants
• Cofactor Expansion
• Properties of Determinants
• Row Reducing for Determinants

### Vector Spaces

• Definition of a Vector Space
• Examples: Polynomials, Arrows, Sequences, Functions, Matrices
• Definition of Subspaces
• Spanning Sets
• Null Space of a Matrix
• Column Space of a Matrix
• Kernel and Range of Transformations
• Basis of a Subspace
• Pivot Columns as Basis
• Unique Representation Theorem Coordinates from a Basis
• Euclidean Coordinates
• Dimension of a Vector Space
• Basis Theorem
• Row Spaces
• Rank Theorem
• Change of Coordinates
• Difference Equations
• Markov Chains

### Eigenvalues and Eigenvectors

• Definition of Eigenvectors
• Triangular Matrices
• Distinct Eigenvalues
• Characteristic Equation
• Similar Matrices
• Diagonalization of Matrices
• Eigenvectors and Transformations
• Complex Eigenvalues
• Predator-Prey Systems

### Orthogonality and Least Squares

• Inner Products
• Length of a Vector
• Distance between Vectors
• Orthogonal Vectors
• The Pythagorean Theorem
• Orthogonal Complement
• Angles and Vectors
• Orthogonal Basis
• Orthogonal Projection
• Orthonormal Sets
• Orthogonal Matrices
• Orthogonal Decomposition
• Geometric View of Projections
• Best Approximation Theorem
• Gram-Schmidt Process
• QR Factorization
• Least Squares Problem
• Solution to Least Squares Problem
• Normal Equations
• Inverse of ATA
• QR Factorization and Least Squares
• Least-Squares Line of Best Fit
• Least-Squares Fitting of Curves

### Symmetric Matrices and Quadratic Forms

• Diagonalization of Symmetric Matrices
• Spectral Theorem
• Change of Variable in a Quadratic Form
• Principal Axes

## System Requirements

This course requires high-speed Internet access (such as Cable or DSL). Your browser will need to allow javascript, login cookies, and popup windows from ctyjhu.org, bluejay.cty.jhu.edu, and any other course web sites.

Students will also need: