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Let's assume that $X$ is a finite dimensional real space with chosen orientation and $Y$ is its subspace with chosen orientation. In what a way to choose orientation in the quotient space $X/Y$?

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This is answered in Example 1.7 of the file at http://math.stanford.edu/~conrad/diffgeomPage/handouts/orient.pdf. A concrete interpretation in terms of bases of $X/Y$ and their lifts to $X$ is given near the end of the example.

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