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Modular Arithmetic: Inverses and Equations

Explores modular arithmetic, emphasizing inverses and equations in Z/mZ, with practical examples and exercises.

Division in Rational Numbers

Explores division in rational numbers and the extension of addition from integers.

Element Inverse for Multiplication

Covers the concept of element inverse for multiplication in complex numbers.

Modular Arithmetic: Introducing Z/mZ

Introduces Z/mZ for writing equations with congruence classes in modular arithmetic.

Mapping Functions and Surjections

Explores mapping functions, surjections, injective and surjective functions, and bijective functions.

Division and Multiplication in Rational Numbers

Explores division and multiplication in rational numbers, emphasizing the concept of multiplying by the inverse.

Group Theory: Basics

Covers the basics of group theory, including definitions, examples, and isometries.

Orthogonality and Eigenvalues

Explores orthogonality, eigenvalues, and diagonalization in linear algebra, focusing on finding orthogonal bases and diagonalizing matrices.

Group Theory: Definitions and Properties

Introduces group theory concepts, including definitions and properties of groups, rings, and fields.

Inverse Element for Multiplication

Covers the inverse element for multiplication and introduces the Euler and Moivre formulas in complex numbers.

Sylvester's Theorem: Orthogonal Bases

Explores Sylvester's Theorem and the importance of orthogonal bases in linear algebra.

Modular Arithmetic: Foundations and Applications

Introduces modular arithmetic, its properties, and applications in cryptography and coding theory.

Limits and Examples

Explores the concept of limits and provides examples of their existence and non-existence in various functions.

Matrix Multiplication and Inverses

Covers matrix product, inverses, and properties of invertible matrices.

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Matrix Inversion: Basics and Properties

Covers the basics of matrix inversion, properties of matrix multiplication, and the uniqueness of matrix inverses.

Differential Equations: Solutions and Uniqueness

Explores differentiability, continuity, solutions, and uniqueness in differential equations, emphasizing well-posed problems and real-world applications.

Commutative Groups: Foundations for Cryptography

Covers commutative groups and their significance in cryptography.

Local Invertibility: Vector Fields

Covers local invertibility of vector fields and the Inverse Function Theorem.

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