Orientation of a finite-dimensional real vector space

Let $V$ be a finite-dimensional real vector space (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). Two ordered bases $\mathcal{B},{\mathcal{B}}^{\mathrm{\prime }}$ of $V$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis) are declared equivalent when the determinant of the unique change-of-basis linear isomorphism $P:V\to V$ carrying $\mathcal{B}$ to ${\mathcal{B}}^{\mathrm{\prime }}$ satisfies $\det P>0$, where the determinant is The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space and positivity is the order of the real field The reals form a totally ordered field.

This is an equivalence relation. The identity change of basis has determinant $1>0$. If $\mathcal{B}\sim {\mathcal{B}}^{\mathrm{\prime }}$ through $P$, then ${\mathcal{B}}^{\mathrm{\prime }}\sim \mathcal{B}$ through $P^{-1}$, and $\det (P^{-1})=(\det P{)}^{-1}>0$. If $\mathcal{B}\sim {\mathcal{B}}^{\mathrm{\prime }}\sim {\mathcal{B}}^{\mathrm{\prime }\mathrm{\prime }}$ through $P,Q$, then $\det (Q\circ P)=\det Q\det P>0$ by For same-sized finite square matrices over a commutative ring, $\det (AB)=\det (A)\det (B)$.

An orientation of $V$ is an equivalence class of ordered bases under this relation; a basis in the chosen class is positively oriented for that orientation. When $\dim V\ge 1$ there are exactly two orientations: fixing one ordered basis ${\mathcal{B}}_0$, every other basis $\mathcal{B}$ has $\det P>0$ or $\det P<0$ by the trichotomy of The reals form a totally ordered field. If $\dim V=1$, replacing the sole basis vector of ${\mathcal{B}}_0$ by its negative produces a basis with determinant $-1$; if $\dim V\ge 2$, interchanging two entries of ${\mathcal{B}}_0$ does the same. Thus both classes occur. When $\dim V=0$ the only ordered basis is the empty one and there is exactly one orientation.

Orientations depend only on the real vector-space structure; no inner product or basis preference enters the definition.