Chapter 1

Rings and Subrings

Definition and examples of rings

medium • 1 min read

Properties of rings

medium • 2 min read

Subrings

medium • 3 min read

Integral domains

medium • 4 min read

Fields and their properties

medium • 5 min read

Chapter 2

Ideals and Quotient Rings

Definition of ideals

medium • 1 min read

Prime and maximal ideals

medium • 2 min read

Quotient rings

medium • 3 min read

Ring homomorphisms

medium • 4 min read

Isomorphism theorems for rings

medium • 5 min read

Chapter 3

Vector Spaces and Subspaces

Definition and examples of vector spaces

medium • 1 min read

Subspaces and criteria

medium • 2 min read

Linear combinations

medium • 3 min read

Span of a set

medium • 4 min read

Linear dependence and independence

medium • 5 min read

Chapter 4

Basis and Dimension

Definition of basis

medium • 1 min read

Dimension of a vector space

medium • 2 min read

Extension to a basis

medium • 3 min read

Coordinates relative to a basis

medium • 4 min read

Change of basis

medium • 5 min read

Chapter 5

Linear Transformations and Matrices

Definition of linear transformations

medium • 1 min read

Null space and range

medium • 2 min read

Rank-nullity theorem

medium • 3 min read

Matrix representation of a linear transformation

medium • 4 min read

Algebra of linear transformations

medium • 5 min read