Algebra begins with numbers and equations, develops methods for working with vectors and linear maps, and then studies the structures behind those operations. Every segment belongs to the same Algebra strand.
Starting with numbers and equations
- Real numbers and algebraic expressions
- Linear equations, inequalities and graphs
- Polynomials, factorisation and quadratic equations
- Functions, exponentials and logarithms
Continue with sequences, finite sums and counting, then complex numbers and polynomial roots. Complex polar forms use trigonometry.
Studying linear algebra
Start with geometric vectors, then use matrices to organise and solve systems. The next steps explain the spaces behind those calculations and the maps between them.
- Vectors and vector operations
- Matrices and linear systems
- Vector spaces and linear maps
- Inner products, orthogonality and least squares
- Determinants, orientation and volume
- Eigenvalues, eigenvectors and diagonalisation
- Spectral theory and matrix decompositions
Basic set notation is background for vector spaces and algebraic structures. Set theory belongs to its own branch; proof methods are practised here alongside the algebra they explain. Complex arithmetic is needed for the full eigenvalue and spectral theory.
For each topic, work through a calculation, explain its geometric or structural meaning, and practise the relevant proof. Move between equations, matrices and linear maps so that the connections stay visible.
Multilinear algebra and tensors extends linear algebra to forms, tensor products and exterior algebra. Numerical algorithms continue in numerical approximation and linear algebra.
Studying algebraic structures
- Groups and symmetry
- Rings, ideals and polynomial structures
- Modules and algebras
- Fields, extensions and Galois theory
- Group representations and characters
- Lie groups and Lie algebras
Groups generalise familiar operations and symmetries. Rings bring addition and multiplication together; modules extend vector spaces to scalars in a ring. Field extensions explain polynomial roots. Representations connect groups to linear algebra, and Lie theory also draws on calculus.
The Algebra path follows one continuous reading sequence through all these segments. Prerequisite lists record the material each segment uses. Connections from other subjects supply background without changing the Algebra sequence.
Books and courses
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OpenStax, Elementary Algebra 2e and Intermediate Algebra 2e: arithmetic, equations, graphs, systems and polynomial work.
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Serge Lang, Introduction to Linear Algebra, second edition: vectors, matrices, vector spaces, linear maps, orthogonality, determinants and eigenvalues.
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Gilbert Strang, Linear Algebra and Its Applications, fourth edition: elimination, vector spaces, orthogonality, determinants, eigenvalues, positive definite matrices and computation. MIT’s 18.06 course provides a related lecture sequence and exercises.
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Thomas W. Judson, Abstract Algebra: Theory and Applications: proof and integer preliminaries, groups, rings, polynomial structures, fields and Galois theory.
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MIT, Introduction to Representation Theory: the connection between linear algebra and algebraic structures.
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OpenStax, College Algebra 2e: functions, polynomial and rational equations, exponential and logarithmic models, systems and finite sequences.
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MIT, Algebra I and Algebra II: group structure, rings, modules, canonical forms and field extensions.
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Netlib, Templates for the Solution of Linear Systems: stationary and Krylov methods, convergence and preconditioning.