Equivalence · Relation · Transformation · Structure

Algebra

Algebra grows by abstraction: from concrete unknowns, to symbolic relations, to linear transformations, to general structures whose rules survive translation.

Equivalence rail

Rewrite without changing what is true.

2(x + 3) = 10

same relationship

An equation states that two expressions represent the same value.

The algebra ladder

The objects become more abstract; preserving relationships remains the connective idea.

concrete → symbolic → linear → structural

Unknowns enter arithmetic

Pre-Algebra

Numbers become placeholders, expressions, and equations.

2(x + 3) = 10

Preserves equality + arithmetic laws

Inside this branch

Integers & Negatives Order of Operations Number Properties Ratios & Proportions Advanced Fractions Exponents Expressions Solving for X

Open Pre-Algebra

Relations become objects

Integrated Algebra

Equations become graphs, functions, systems, and families of relationships.

f(x) = y

Preserves solutions + functional relationships

Inside this branch

Algebra Fundamentals Graphing Linear Equations Systems of Equations Algebraic Inequalities Quadratic Equations Factoring Exponents Radical Expressions Functions Rational Expressions Complex Numbers

Open Integrated Algebra

Equations become geometry

Linear Algebra

Coordinates, vectors, and matrices turn systems into transformations of space.

A x = b

Preserves linear combinations + vector structure

Inside this branch

Vectors Matrices Linear Systems Determinants Linear Transformations Vector Spaces Eigen Theory Orthogonality Singular Value Decomposition

Open Linear Algebra

Rules become structures

Abstract Algebra

Algebra studies the operations themselves and the structures that survive translation.

φ(ab) = φ(a)φ(b)

Preserves operations + algebraic structure

Inside this branch

Groups Maps

Ringsplanned

Fields

Vector Spacesplanned

Galois Theoryplanned

Open Abstract Algebra