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  "body": "\n## Definition\n\nLet $V$ be a real vector space of finite dimension $n \\ge 1$.\nTwo ordered [[bases|0KCQ9Y3QKA]] $B$ and $B'$ of $V$ have the *same orientation* when the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$ has positive [[determinant|00KDZRW4C0]].\nThis is an equivalence relation with exactly two classes.\nAn *orientation* of $V$ is a choice of one of these two classes.\n\n## Remarks\n\nA vector space with a chosen orientation is *oriented*, and the bases in the chosen class are *positively oriented*.\nThe standard orientation of $\\R^n$ is the class of the standard basis.\nIn $\\R^2$, exchanging the two basis vectors reverses the orientation.\nAn invertible linear map keeps the orientation if and only if its determinant is positive.\n\n## Explanations\n\n### L1 human\n\nAn orientation fixes what counts as turning to the left, or as a right-handed system.\nIn the plane, a counterclockwise pair of basis vectors and a clockwise pair have opposite orientations.\nA mirror reflection changes the orientation, and a rotation keeps it.\n",
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