# Linear classical group

A group of non-singular linear transformations of a finite-dimensional vector space $ E $
over a skew-field $ K $
that is a classical group (see also Linear group). The most important types of linear classical groups are the following: the general linear group $ \mathop{\rm GL} _ {n} ( K) $,
the special linear group $ \mathop{\rm SL} _ {n} ( K) $
and the unitary group $ U _ {n} ( K , f ) $(
where $ n = \mathop{\rm dim} E $
and $ f $
is a Hermitian or skew-Hermitian form on $ E $,
relative to an involution of $ K $).
When $ K $
is also commutative, special important cases are: the symplectic group $ \mathop{\rm Sp} _ {n} ( K) $
and the orthogonal group $ O _ {n} ( K , f ) $(
$ f $
a quadratic form on $ E $
and $ K $
of characteristic not 2).

**How to Cite This Entry:**

Linear classical group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Linear_classical_group&oldid=47649