Introduction

Introduction

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Data Structures for Linear Algebra

What Linear Algebra Is

Plotting a System of Linear Equations

Linear Algebra Exercise

Tensors

Scalars

Vectors and Vector Transposition

Norms and Unit Vectors

Basis, Orthogonal, and Orthonormal Vectors

Matrix Tensors

Generic Tensor Notation

Exercises on Algebra Data Structures

Tensor Operations

Segment Intro

Tensor Transposition

Basic Tensor Arithmetic, incl. the Hadamard Product

Tensor Reduction

The Dot Product

Exercises on Tensor Operations

Solving Linear Systems with Substitution

Solving Linear Systems with Elimination

009 Visualizing Linear Systems

Matrix Properties

Segment Intro

The Frobenius Norm

Matrix Multiplication

Symmetric and Identity Matrices

Matrix Multiplication Exercises

Matrix Inversion

Diagonal Matrices

Orthogonal Matrices

Orthogonal Matrix Exercises

Eigenvectors and Eigenvalues

Segment Intro

Applying Matrices

Affine Transformations

Eigenvectors and Eigenvalues

Matrix Determinants

Determinants of Larger Matrices

Determinant Exercises

Determinants and Eigenvalues

Eigendecomposition

Eigenvector and Eigenvalue Applications

Matrix Operations for Machine Learning

Segment Intro

Singular Value Decomposition

Data Compression with SVD

The Moore-Penrose Pseudoinverse

Regression with the Pseudoinverse

The Trace Operator

Principal Component Analysis (PCA)

Resources for Further Study of Linear Algebra

Limits

Segment Intro

Intro to Differential Calculus

Intro to Integral Calculus

The Method of Exhaustion

Calculus of the Infinitesimals

Calculus Applications

Calculating Limits

Exercises on Limits

Derivatives and Differentiation

Segment Intro

The Delta Method

How Derivatives Arise from Limits

Derivative Notation

The Derivative of a Constant

The Power Rule

The Constant Multiple Rule

The Sum Rule

Exercises on Derivative Rules

The Product Rule

The Quotient Rule

The Chain Rule

Advanced Exercises on Derivative Rules

The Power Rule on a Function Chain

Automatic Differentiation

Segment Intro

What Automatic Differentiation Is

Autodiff with PyTorch

Autodiff with TensorFlow

The Line Equation as a Tensor Graph

Machine Learning with Autodiff

Partial Derivative Calculus

Segment Intro

What Partial Derivatives Are

Partial Derivative Exercises

Calculating Partial Derivatives with Autodiff

Advanced Partial Derivatives

Advanced Partial-Derivative Exercises

Partial Derivative Notation

The Chain Rule for Partial Derivatives

Exercises on the Multivariate Chain Rule

Point-by-Point Regression

The Gradient of Quadratic Cost

Descending the Gradient of Cost

The Gradient of Mean Squared Error

Backpropagation

Higher-Order Partial Derivatives

Exercise on Higher-Order Partial Derivatives

Integral Calculus

Segment Intro

Binary Classification

The Confusion Matrix

The Receiver-Operating Characteristic (ROC) Curve

What Integral Calculus Is

The Integral Calculus Rules

Indefinite Integral Exercises

Definite Integrals

Numeric Integration with Python

Definite Integral Exercise

Finding the Area Under the ROC Curve

Resources for the Further Study of Calculus

What Next

More Lectures are on their Way!