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Here, an "affine set" may be defined as the translation of a vector subspace.
en.wikipedia.org
The degrees-of-freedom, here a parameter of the distribution, can still be interpreted as the dimension of an underlying vector subspace.
en.wikipedia.org
In order to do this, we have to translate the geometrical tangent space to so that it passes through the origin rather than, and hence defines a -dimensional vector subspace.
en.wikipedia.org

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