 Orientation-preserving general linear group

The real invertible matrices with positive determinant form a subgroup of the general linear group. The identity has determinant one, multiplicativity of the determinant preserves positivity under products, and $\det (A^{-1})=1/\det A>0$ preserves it under inversion. More precisely this is the kernel of a group homomorphism $A\mapsto \mathrm{sgn}(\det A)$ to $\{1,-1\}$, so it is a normal subgroup of index two for $n\ge 1$. Its elements preserve an orientation of a vector space.