Vector spaceAlgebraic structure in linear algebraVector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
Scalar (mathematics)In mathematics, more specifically in linear algebra, a scalar is an element of a field which is used to define a vector space through the operation of scalar multiplication: a vector (denoted v) multiplied by a scalar (denoted a) produces another vector (av). Real numbers and complex numbers may be used as scalars in real and complex vector spaces, respectively.
Commutative ringIn mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
Field (mathematics)Algebraic structure with addition, multiplication, and divisionA field is an algebraic structure that is closed under the four usual arithmetic operations.In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics.Ring (mathematics)Algebraic structure with addition and multiplicationIn mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative.
Abelian groupIn mathematics, an abelian group,[note 1] also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples.
Group (mathematics)In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.
Representation theoryRepresentation theory is a branch of mathematics that studies abstractalgebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras.
Distributive propertyIn mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x⋅(y+z)=x⋅y+x⋅z{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} is always true in elementary algebra. For example, in elementary arithmetic, one has 2⋅(1+3)=(2⋅1)+(2⋅3).{\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).} Therefore, one would say that multiplicationdistributes over addition.Semigroup actionIn algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set.
IntegerAn integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.*As an Amazon Associate I earn from qualifying purchases.