# Portal:Algebra

## Algebra

**Algebra** is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muḥammad ibn Mūsā al-Khwārizmī titled *Kitab al-Jabr al-Muqabala* (meaning "*The Compendious Book on Calculation by Completion and Balancing*"), which provided operations for the systematic solution of linear and quadratic equations.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

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In this shear transformation of the Mona Lisa, the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an eigenvector of this particular transformation and the blue vector is not. |

In mathematics, an **eigenvector** of a transformation is a vector which that transformation simply multiplies by a constant factor, called the **eigenvalue** of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The **eigenspace** for a factor is the set of eigenvectors with that factor as eigenvalue.

In the specific case of linear algebra, the *eigenvalue problem* is this: given an *n* by *n* matrix *A*,what nonzero vectors *x* in exist, such that *Ax* is a scalar multiple of *x*?

The scalar multiple is denoted by the Greek letter *λ* and is called an *eigenvalue* of the matrix A, while *x* is called the *eigenvector* of *A* corresponding to *λ*. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.

It is common to prefix any natural name for the vector with *eigen* instead of saying *eigenvector*. For example, *eigenfunction* if the eigenvector is a function, *eigenmode* if the eigenvector is a harmonic mode, *eigenstate* if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. *eigenfrequency* if the eigenvalue is (or determines) a frequency.

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These are all the connected **Dynkin diagrams**, which classify the irreducible root systems, which themselves classify simple complex Lie algebras and simple complex Lie groups. These diagrams are therefore fundamental throughout Lie group theory.

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## Did you know?

- ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
- ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
- ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
- ...that the classification of finite simple groups was not completed until the mid 1980s?
- ...that a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers?
- ...that the classification of finite simple groups is of more than 10000 pages.

*Topics in algebra*

General | Elementary algebra | Key concepts | Linear algebra |
---|---|---|---|

Algebraic structures | Groups | Rings and Fields | Other |

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