Professor Biography
Course Scope
Lecture 1—Linear Algebra: Powerful Transformations
Transformations
Lecture 2—Vectors: Describing Space and Motion
Vectors
Linear Combinations
Abstract Vector Spaces
Lecture 3—Linear Geometry: Dots and Crosses
The Dot Product
Properties of the Dot Product
A Geometric Formula for the Dot Product
The Cross Product
Describing Lines
Describing Planes
Lecture 4—Matrix Operations
What Is a Matrix?
Matrix Multiplication
The Identity Matrix
Other Matrix Properties
Lecture 5—Linear Transformations
Multivariable Functions
Definition of a Linear Transformation
Properties of Linear Transformations
Matrix Multiplication Is a Linear Transformation
Examples of Linear Transformations
Lecture 6—Systems of Linear Equations
Linear Equations
Systems of Linear Equations
Solving Systems of Linear Equations
Gaussian Elimination
Getting Infinitely Many or No Solutions
Quiz for Lectures 1–6
Lecture 7—Reduced Row Echelon Form
Reduced Row Echelon Form
Using the RREF to Find the Set of Solutions
Row-Equivalent-Matrices
Lecture 8—Span and Linear Dependence
The Span of a Set of Vectors
When Is a Vector in the Span of a Set of Vectors?
Linear Dependence of a Set of Vectors
Linear Independence of a Set of Vectors
Lecture 9—Subspaces: Special Subsets to Look For
The Null-Space of a Matrix
Subspaces
The Row Space and Column Space of a Matrix
Lecture 10—Bases: Basic Building Blocks
Geometric Interpretation of Row, Column, and Null-Spaces
The Basis of a Subspace
How to Find a Basis for a Column Space
How to Find a Basis for a Row Space
How to Find a Basis for a Null-Space
The Rank-Nullity Theorem
Lecture 11—Invertible Matrices: Undoing What You Did
The Inverse of a Matrix
Finding the Inverse of a 2 × 2 Matrix
Properties of Inverses
Lecture 12—The Invertible Matrix Theorem
The Importance of Invertible Matrices
Finding the Inverse of an n × n Matrix
Criteria for Telling If a Matrix Is Invertible
Quiz for Lectures 7–12
Lecture 13—Determinants: Numbers That Say a Lot
The 1 × 1 and 2 × 2 Determinants
The 3 × 3 Determinant
The n × n Determinant
Calculating Determinants Quickly
The Geometric Meaning of the n × n Determinant
Consequences
Lecture 14—Eigenstuff: Revealing Hidden Structure
Population Dynamics Application
Understanding Matrix Powers
Eigenvectors and Eigenvalues
Solving the Eigenvector Equation
Return to Population Dynamics Application
Lecture 15—Eigenvectors and Eigenvalues: Geometry
The Geometry of Eigenvectors and Eigenvalues
Verifying That a Vector Is an Eigenvector
Finding Eigenvectors and Eigenvalues
Matrix Powers
Lecture 16—Diagonalizability
Change of Basis
Eigenvalues and the Determinant
Algebraic Multiplicity and Geometric Multiplicity
Diagonalizability
Computing Matrix Powers
Similar Matrices
Lecture 17—Population Dynamics: Foxes and Rabbits
Recalling the Population Dynamics Model
High Predation
Low Predation
Medium Predation
Lecture 18—Differential Equations: New Applications
Solving a System of Differential Equations
Complex Eigenvalues
Quiz for Lectures 13–18
Lecture 19—Orthogonality: Squaring Things Up
Orthogonal Sets
Orthogonal Matrices
Properties of Orthogonal Matrices
The Gram-Schmidt Process
QR-Factorization
Orthogonal Diagonalization
Lecture 20—Markov Chains: Hopping Around
Markov Chains
Economic Mobility
Theorems about Markov Chains
Lecture 21—Multivariable Calculus: Derivative Matrix
Single-Variable Calculus
Multivariable Functions
Differentiability
The Derivative
Chain Rule
Lecture 22—Multilinear Regression: Least Squares
Linear Regression
Multiple Linear Regression
Invertibility of the Gram Matrix
How Good Is the Fit?
Polynomial Regression
Lecture 23—Singular Value Decomposition: So Cool
The Singular Value Decomposition
The Geometric Meaning of the SVD
Computing the SVD
Lecture 24—General Vector Spaces: More to Explore
Functions as Vectors
General Vector Spaces
Fibonacci-Type Sequences as a Vector Space
Space of Functions as Vector Spaces
Solutions of Differential Equations
Ideas of Fourier Analysis
Quiz for Lectures 19–24
Solutions
Lectures 1–6
Lectures 7–12
Lectures 13–18
Lectures 19–24
Bibliography