Suppose ## V ## is a vector space and ## W ## is a subspace of ## V ##. Then for ## |a\rangle, |b\rangle \in V ##, say that ## |a\rangle ## and ## |b\rangle ## are equivalent (in the sense of an equivalence relation), if ## |a\rangle - |b\rangle \in W ##. Denote the equivalence class of ## |a\rangle ## by ## [\![ a ]\!] ##. The factor space (also called the quotient space) is defined via the underlying set ##\{ [\![ a ]\!] \; | \; | a\rangle \in V \} ## together with the addition rule ## \alpha [\![ a ]\!] + \beta [\![ b ]\!] = [\![ \alpha a + \beta b ]\!] ## for combining elements in this set, with ##\alpha ## and ##\beta## scalars of the original space ## V ##.(adsbygoogle = window.adsbygoogle || []).push({});

For an intuitive idea of the meaning of a factor space, let ## V = \mathbb{R}^2 ##, ## W ## be a line passing through the origin, and ## | v \rangle ## be a position vector originating from the origin. Then ## [\![ v ]\!] ## is the line obtained by shifting every point in ## W ## (regarded as a position vector) by ## |v\rangle ##. The factor space is the set of all lines that are parallel to ## W ##.

Does anyone know of a good example, from physics, of a factor space?

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# Example of a factor space in physics

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