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Thread: Affine space description

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    Forum Admin topsquark's Avatar
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    Affine space description

    Okay, it's time to be getting back at some Differential Geometry. Before I start I would like a favor: Can someone give me a simple description of what an affine space is? I've gotten the impression it's essentially a vector space where the ends of the vector can be any place we desire unlike Euclidean space where the ends of the vectors are nailed down to a given origin. Am I at least close?

    Thanks!

    -Dan
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    Re: Affine space description

    Some what. In a vector space, we have the 0 vector as well as the condition that if u and v are in the space then u+ v and av, for v any scalar, are in the space. An "affine" space doesn't have that property but- given any affine space, if u and v are in the space then there exist a vector, w, such u= w+ x, v= w+ y, x+ y+ w is in the affine space, ax+ w is in the space, and at+ w is in the space.

    Geometrically, you can think of a one dimensional vector space as a line through the origin or a two dimensional vector space as a plane through the origin in three space. You can think of an "affine" space as a line or plane that does NOT contain the origin.
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    Forum Admin topsquark's Avatar
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    Re: Affine space description

    Thank you. That clears up a lot of things for me.

    -Dan
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