special linear group

Given a vector space $V$ , the special linear group $\SL(V)$ is defined to be the subgroup of the general linear group $\operatorname{GL}(V)$ consisting of all invertible linear transformations $T: V \longrightarrow V$ in $\operatorname{GL}(V)$ that have determinant 1.

If $V = \mathbb{F}^n$ for some field $\mathbb{F}$ , then the group $\SL(V)$ is often denoted $\SL(n,\mathbb{F})$ or $\SL_n(\mathbb{F})$ , and if one identifies each linear transformation with its matrix with respect to the standard basis, then $\SL(n,\mathbb{F})$ consists of all $n \times n$ matrices with entries in $\mathbb{F}$ that have determinant 1.