Whitehead's lemma is a technical result in abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 used in algebraic K-theory

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

. It states that a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 of the form


is equivalent to the identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

 by elementary transformations (that is, transvections):


Here, indicates a matrix whose diagonal block is and entry is .

The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group

General linear group

In mathematics, the general linear group of degree n is the set of n×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...

 is the group generated by elementary matrices. In symbols,.

This holds for the stable group (the direct limit

Direct limit

In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for

one has: