In mathematics, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that L has a linear representation ρ over K, on a finite-dimensional vector space V, that is a faithful representation, making L isomorphic to a subalgebra of the endomorphisms of V. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In abstract algebra, 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. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...

While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group G, it shows not that G has a faithful linear representation (which is not a valid theorem), but that G always has a linear representation that is a local isomorphism with a linear group. It was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov. In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... Kazan State University is located in Kazan, Tatarstan, Russia. ... Nikolai Chebotaryov (1894-1947) was a noted Ukrainian mathematician. ...