[[Linear algebra MOC]]
# Oriented vector space

An **oriented vector space** $(V,\mathbb{K},\sgn)$ is a finite-[[Dimension of a vector space|dimensional]] [[vector space]] $(V, \mathbb{K})$ with a fixed choice of which [[Ordered basis|ordered bases]] are positively oriented. #m/def/linalg 
Two ordered bases $\mathcal{B} = (\vab e_{i})_{i=1}^n$ and $\mathcal{B}' = (\vab e_{i}')_{i=1}^n$ of a vector space $V$ have the same orientation iff the unique [[Linear map|linear automorphism]] $A : \mathcal{B} \mapsto \mathcal{B}'$ has a positive [[Matrix determinant|determinant]].
This divides the possible bases of $V$ into two equivalence classes.
An orientation of $V$ is thus a choice of which of these equivalence classes is positive and which is negative.

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#state/tidy | #lang/en | #SemBr