# 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).