Linear Algebra

Introduction

Why Linear Algebra and Course Overview
Discussion of Essence of Linear Algebra Chapters 1-6
Intro to Set Theory, Quantifiers, and Proofs
Discussion of Essence of Linear Algebra Chapters 7-9 and 13-15

Vector Spaces and Linear Maps

Vector Spaces and Intro to Axler
Subspaces, Direct Sums, Span, and Linear Independence
Markov Chains Guest Lecture
Basis and Dimension
Linear Maps, Null Space, and Range

Matrices and Eigenvectors

Matrices and Inverses
Gaussian Elimination
Polynomials
Eigenvalues and Eigenvectors
Quotient Spaces
Eigenspaces and Diagonalization

Inner Product Spaces and the Spectral Theorem

Inner Product Spaces
Gram-Schmidt and Minimization
Adjoints and the Spectral Theorem
Positive Operators, Isometries, and the Polar Decomposition

Additional Topics

Generalized Eigenvectors
Characteristic Polynomials and Jordan Form
Recommender Systems Guest Lecture
Determinants

Higher Math Beyond Linear Algebra

Higher Math Beyond Linear Algebra I
Higher Math Beyond Linear Algebra II

The Birth of the Course

Teaching Linear Algebra in the 21st Century

Differential Equations

Verification

Introduction
Example

Direction Fields

Introduction
The Method of Isoclines
The Phase Line

First Order DEs: Separable

Introduction
Example 1
Example 2

First Order DEs: Linear

Introduction
Example
Mixing Problem 1
Mixing Problem 2
Mixing Problem 3

First Order DEs: Exact

Introduction 1
Introduction 2
Example 1
Example 2

Second Order DEs: Homogeneous

Linearity
Linear Independence
Homogeneous Introduction
Characteristic Equation: Distinct Roots
Characteristic Equation: A Double Root
Characteristic Equation: Complex Roots
Real Roots Practice
Complex Roots Practice

The Laplace Transform

Introduction
Laplace Transform of Exponentials
Laplace Transform of Sines and Cosines
Laplace Transform of Polynomials
Laplace Transform of a Derivative
The Inverse Laplace Transform
Linearity of the Inverse Laplace Transform
Solving DEs
Solving DEs: Translation
Solving DEs: Partial Fraction Decomposition

AP Practice

Worked AP Calculus BC Full Exams

2016 Multiple Choice (no calculator)
2016 Multiple Choice (calculator)
2017 Free Response (calculator)
2017 Free Response (no calculator)
2013 Multiple Choice (no calculator)

Worked AP Calculus BC Specific Questions

2012 FRQ 5 (parts A and B)
2012 FRQ 5 (part C)
2009 FRQ 4 (part A)
2009 FRQ 4 (parts B and C)

Worked AP Physics C: E&M FRQ Sections

FRQ Practice | 2013 AP Physics C: E&M
FRQ Practice | 2018 AP Physics C: E&M

Worked AP Chemistry FRQ Sections

FRQ Practice | 2017 AP Chemistry

Trigonometry

What is Trigonometry?

Introduction
Exploration
Introduction and Pythagorean Identity
Even-Odd Identities

Proofs of Identities

The Double Angle Identities
The Sum identities
Half Angle and Angle Subtraction Identities

Fun

Programs

Sandbox Game
Gravity Crash

Self Study

Strategies for Math
Map of Mathematics
Linear Algebra
Multivariable Calculus
Abstract Algebra
Galois Theory

Miscellaneous

Children's Book Narrated: The Magic Seed
Exchange Networks
How did Change Accelerate?

The Mandelbrot Set

Introduction
Example
Computer-Rendered

The Binomial Theorem

Part 1
Part 2

Number Theory

The Sum of All Terminating Fractions
The Sum of the Factors of 11!
Factors of 11! satisfying gcd(d, 396) = 12

The Best of Math

Derivation of the Sum of a Geometric Series
Derivation of the Quadratic Equation
Intuition for Linear Programming
The Area under the Bell Curve
The Rotation Matrix
How are Circumference and Area Connected?
The Surface Area and Volume of Spheres
Projectile Motion?
Proof of the Dot Product Theorem
The Reflection Matrix
The Corner Cube Problem
Why 68% of Data Falls within 1 Standard Deviation
Defining the Cross Product from Scratch
Circles are Infinity-Sided Polygons

US History

Class Notes

US History 2016-17 Notes

Tools

General

Flashcards
Function Generator
Rebel Reader (Tool)
Rebel Reader (Video)

Linear Algebra

Function Fudger

About

Arongil Academy is a free, online library of mathematics.

Contact

You can contact Arongil Academy at [email protected].