Linear Algebra: Linear Transformations & Vector Spaces

Course Curriculum

Chapter 1: Linear Transformations & Bases

Lecture 1: What Is A Linear Transformation?

Lecture 2: Projections As Linear Transformations

Lecture 3: Proof That A Projection Is A Linear Transformation

Lecture 4: Linear Transformations: Rotations

Lecture 5: Revisiting Rotations

Lecture 6: Linear Transformations: A Non-Example

Lecture 7: Linear Transformations As Matrix-Vector Product

Lecture 8: Constructing A Projection Matrix

Lecture 9: What Is A Basis?

Lecture 10: Linear Transformations: Transforming Basis Vectors

Lecture 11: Building A Linear Transformation Matrix

Lecture 12: Determinants As Linear Transformations

Lecture 13: When The Determinant of A Transformation Matrix is 0

Chapter 2: Linear Dependence, Span, Rank, & Vector Space Concepts

Lecture 1: Linear Independence: An Introduction

Lecture 2: Testing For Linear Independence

Lecture 3: Testing For Linear Independence Using Determinant

Lecture 4: Eyeballing Linear Dependence

Lecture 5: What Is Span?

Lecture 6: Defining Rank

Lecture 7: Computing Rank: Concrete Example

Lecture 8: Proof That Column Rank = Row Rank (Part 1)

Lecture 9: Proof That Column Rank = Row Rank (Part 2)

Lecture 10: Column Space Definition

Lecture 11: Row Space Definition

Lecture 12: Null Space Definition

Chapter 3: Linear Subspaces

Lecture 1: What Is A Linear Subspace?

Lecture 2: Basis of A Subspace

Lecture 3: Linear Subspaces: Multiple Bases

Lecture 4: Changing Basis

Lecture 5: Generalizing Change of Basis

Lecture 6: Inverting The Change of Basis Matrix

Lecture 7: Transformation Matrix in Different Basis

Lecture 8: Transformation Matrix in Different Basis: Example (Part 1)

Lecture 9: Transformation Matrix in Different Basis: Example (Part 2)

Lecture 10: Getting Back To Standard Transformation Matrix

Lecture 11: Changing Basis & Transformations: A Review

Lecture 12: Changing Basis To Find Transformation Matrix: Formulaic Approach

Lecture 13: Orthonormal Basis: Definition

Lecture 14: Gram Schmidt & Orthonormal Basis

Chapter 4: Eigenvalues & Eigenvectors

Lecture 1: Introduction to Eigenvalues & Eigenvectors

Lecture 2: Finding Eigenvalues – 2×2 Example

Lecture 3: Solving Eigenvalues: Explaining Why

Lecture 4: Finding Eigenvectors: 2×2 Example

Lecture 5: Visualizing Eigenvectors

Lecture 6: Finding Eigenvalues of 3×3 Matrix

Lecture 7: Finding Eigenvectors of 3×3 Matrix

Lecture 8: Checking Our Work

Lecture 9: Diagonalization

Lecture 10: What Is An Eigenbasis?

Chapter 5: Abstract Vector Spaces

Lecture 1: Defining An Abstract Vector Space

Lecture 2: Viewing Functions As Vectors

Lecture 3: Abstract Vectors Spaces: A Bizarre Example