AutomorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.
Group isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.
Linear mapMathematical function, in linear algebraIn mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.
General linear groupIn mathematics, the general linear group of degree n {\displaystyle n} is the set of n × n {\displaystyle n\times n} invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
Category theoryCategory theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories.
MorphismIn mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not be maps, but they can be composed in a way that is similar to function composition. Morphisms and objects are constituents of a category.Symmetry groupIn group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).
Dimension (vector space)In mathematics, the dimension of a vector spaceV is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined.
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.
Group (mathematics)Set with associative invertible operationThe manipulations of the Rubik's Cube form the Rubik's Cube group.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. 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.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.Function compositionIn mathematics, the composition operator ∘ {\displaystyle \circ } takes two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘ g {\displaystyle f\circ g} (pronounced " f {\displaystyle f} of g {\displaystyle g} ") is evaluated at an input x {\displaystyle x} , the result is ( f ∘ g ) ( x ) = f ( g ( x ) ) {\displaystyle (f\circ g)(x)=f(g(x))} .*As an Amazon Associate I earn from qualifying purchases.