Algebra is a major branch of pure mathematics and is concerned with the study of the rules of relations and operations and the constructions as well as concepts arising from them, including terms, polynomials, equations and algebraic structures. It has been divided in two subjection, Elementary algebra and complex algebra. Elementary algebra introduces the concept of variables representing numbers and is studied in secondary education. In algebra, statements based on variables are manipulated using the rules of applicable operations like addition, subtraction, etc. to solve the equation including other reasons.
Algebra has its root in ancient Greek mathematics. They created geometric algebra to study the situations of objects. First authentic work on algebra is a book series named Arithmetica written by Diophantus (300 AD) who is identified as “the father of algebra”. In these books, Diophantus described various solutions to solve algebraic equation. Later it was developed by Arabic/Islamic mathematicians. In current history algebra was grown in Europe and become an essential branch of mathematics and engineering. Algebra can be sub-sectioned in following categories:
- Elementary algebra
- Abstract algebra or modern algebra
- Linear algebra
- Matrix algebra
- Universal algebra
Elementary algebra deals with the properties of operations on the real number system, their variables/constants the rules of mathematical expressions and equation. Abstract algebra is concerned with groups, rings and fields while the Linear algebra is a study of vector spaces and their properties. Besides this, the term algebra is also used to indicate various algebraic structures like algebra over a ring, algebra over a set, Boolean algebra, F-algebra and F-coalgebra, relational algebra, sigma algebra and T-algebras. Algebra has a vast range which encompasses various topics of mathematics such as Linear equations, Determinant, Cauchy–Binet formula, Cramer’s rule, Gaussian elimination, Strassen algorithm, Matrix theory, operators of matrix (addition, subtraction, multiplication, transformation ), Characteristic polynomial, Eigenvalue, eigenvector and eigenspace, Cayley–Hamilton theorem, Matrix inversion, invertible matrix, Symmetric matrix, Congruence relation, Hermitian matrix, Antihermitian, Perron–Frobenius theorem, List of matrices, Matrix decompositions (including Cholesky, LU, QR, Singular, Higher-order singular value, Schur), Spectral theorem, Computations, Vectors, Scalar multiplication, Linear combination, Linear span, Linear independence, Coordinates vector, Multilinear algebra, Tensor and treatment of tensors (classical, intermediate and component-free treatment), Tensor algebra, Exterior algebra, Symmetric algebra, Clifford algebra, Geometric algebra, transformation, Projective space and geometry.