Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory.
It states that a matrix of the form is equivalent to the identity matrix by elementary transformations (that is, transvections): Here,
The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group is the group generated by elementary matrices.
[1][2] In symbols, This holds for the stable group (the direct limit 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: where Alt(3) and Sym(3) denote the alternating resp.