Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

Question

I am trying to understand the exact relation between all these things:

• point
• vector
• affine space
• vector space
• basis
• frame
• coordinate system

Can you explain me rigorously (in the mathematical sense) all the inter-relations existing between these geometrical concepts?

Additionnally, a strange thing is that this french wikipedia article seems to define a frame as a basis + origin whereas this english wikipedia article seems to define the relation between a frame and a basis completely differently. So could you explain me the right relation between a basis and a frame (and why one of the article may be erroneous).

Answer 1

An affine space consists of points. The associated vector space contains the translations as mappings of the point space. So in a certain sense, the affine space is a representation of the additive group part of the vector space. The multiplicative part of the vector space then allows to construct lines and segments in the affine space.

For a frame you have to distinguish between linear algebra and differential geometry, where a moving frame is indeed a basis that may depend on the point in some smooth fashion. For instance along a curve the tangent, direction to center of curvature and the cross product of both form a moving frame, or on a surface a basis of the tangent space and the normal.

But I think you are interested in frames in linear algebra and functional analysis. In finite dimension, any basis has that many elements and if you take a generating set with more elements, you have non-trivial linear combinations of the null vector. In infinite dimensions, that relationship breaks down, you can have generating sets of Hilbert spaces that do not form a basis, and that also can not be reduced to a basis by taking a subset of it. That is where the definition of a frame comes in. As a geometric picture, in a frame there is a minimal positive angle between the frame vectors (all signs, or the lines through them). For further details like frame constants and dual frames see wikipedia on frames of vector spaces.