4.2 Linear Algebra with Applications
Delve into the abstract depths of linear algebra: vector spaces, determinants, eigenvalues, wedge products, and more.
What is a Vector?
Waves as Abstract Vectors
Why Vector Spaces?
The Gauss-Jordan Process I
The Gauss-Jordan Process II
Application: Markov Chains I
Real Euclidean Space I
Real Euclidean Space II
Span & Subspaces
Coordinates & Bases
Matrix Subspaces
Application: Coding Theory
Application: Graph Theory I
What Is a Matrix?
Linear Transformations
Matrix Products
Matrix Inverses
Application: Image Compression I
Application: Cryptography
Bivectors
Trivectors & Determinants
Determinant Properties
Multivector Geometry
Dual Space
Tensors & Forms
Tensor Products
Wedges & Determinants
Application: Markov Chains II
Eigenvalues & Eigenvectors
Diagonalizability
Normal Matrices
Jordan Normal Form
Application: Graph Theory II
Application: Discrete Cat Map
Application: Arnold's Cat Map
Inner Product Spaces
Gram-Schmidt Process
Least Squares Regression
Singular Values & Vectors
Singular Value Decompositions
SVD Applications
Course description
Linear algebra plays a crucial role in many branches of applied science and pure mathematics. This course covers the core ideas of linear algebra and provides a solid foundation for future learning. Using geometric intuition as a starting point, the course journeys into the abstract aspects of linear algebra that make it so widely applicable. By the end you'll know about vector spaces, linear transformations, determinants, eigenvalues & eigenvectors, tensor & wedge products, and much more. The course also includes applications quizzes with topics drawn from such diverse areas as image compression, cryptography, error coding, chaos theory, and probability.
Topics covered
- Bases
- Determinants
- Diagonalizable Matrices
- Eigenvalues and Eigenvectors
- Gaussian Elimination
- Gram-Schmidt Process
- Inner Products
- Inverses
- Linear Independence
- Linear Transformations
- Matrices
- Subspaces
- Tensors & Tensor Products
- Vector Spaces
Prerequisites and next steps
A basic understanding of calculus and linear equations is necessary.
Prerequisites
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5.1 Vector Calculus
Extend multivariable calculus to vector fields, then apply your new skills by exploring Maxwell's equations.
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