In mathematics, a faithful representation ρ of a groupG on a vector spaceV is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In more abstract language, this means that the group homomorphism Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
ρ: G → GL(V)
is injective. See also faithful module. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... The introduction of this article does not provide enough context for readers unfamiliar with the subject. ...
Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces.
Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which is a very well-understood theory.
A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V.
In mathematics, the adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation.
The representation is equivalent to that given by the action of G by linear substitution on the space of binary (i.e., 2 variable) quadratic forms.
According to the philosophy in representation theory known as the orbit method, the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits.
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