Part 1 - Linear Algebra

Introduction to Linear Algebra

Introduction, Definition

Linear Combination and Linear Equation

Linear Independence

System of Linear Equations

Geometry of Linear Equations

Solution of System of Linear Equations
1) Geometrical Method
2) Substitution Method
3) Elimination Method

Vectors

Introduction and Geometry

Notation

Basic Properties and Operations
Scalar Multiplication, Addition, Subtraction, Angle

Linear Combination of Vectors

Magnitude of a Vector

Dot Product
Comutation and Geometric Interpretation

Dot Product

Application of Dot Product
Magnitude, Angle Between Vectors, Orthogonality, Norm, Distance

Cross Product
Comutation and Geometric Interpretation

Application of Cross Product

Matrices

System of Linear Equations and Matrix Notation

Application
Images as Multi-dimensional Arrays

Basic Properties and Operations
Images as Multi-dimensional Arrays

Linear Transformation and Matrices

Basic Properties and Operations
Scalar Multiplication, Addition, Subtraction, Transpose

Matrix Product as Composition of Transformations

Types of Matrices
Identity, Orthogonal, Sparse, Triangular

Determinant
Computation and Geometric Interpretation

Inverse of a Matrix
Comutation and Geometric Interpretation

Solution of System of Linear Equations
1) Cramer's Rule 2) Matrix Inverse
3) Echelon Form 4) Reduced Row Echelon Form

Types of Solution
Unique, Infinite, Inconsistent

Eigenvalues

Eigenvectors

Applications of Matrices
Application 1
Application 2

Part 2 - Probability and Random Variables

Probability

Historical Background

Random Experiment

Sample Space, Event

Set Theory

Law of Large Numbers

Axiomatic Definition of Probability

Conditional Probability

Joint Probability

Marginal Probability

Multiplication Rule and Tree Diagrams

Bayes Theorem

Random Variables

Discrete Random Variable

Concept of Random Variable

Discrete Probability Distributions
Binomial, Poisson, Hypergeometric, Bernoulli

Continuous Random Variable

Continuous Probability Distributions
Normal, Exponential

Part 3 - Statistics and Hypothesis Testing

Descriptive Statistics

What is Statistics?

Applicatons of Statistics

Terminologies

Measures of Location

Measures of Variability

Histograms

Measures of Association b/w variables

Inferential Statistics

Hypothesis Tests
1) Null and Alternative Hypothesis 2) Type I and Type II Errors
3) One Tailed and Two Tailed Tests

Inferences about population mean
Variance known and unknown

Inferences about population proportion
Variance known and unknown

ANOVA

Regression

Linear Regression

Polynomial Regression

Goodness of fit

Part 4 - Calculus

Derivatives

Derivative definition and meaning

Highe Order Derivatives

Chain Rule

Optimization

First Derivative Test

Second Derivative Test

Applications

Partial Derivatives

Definition and interpretation

Gradients

Gradient Descent