Affine / vector space

roger
Hi

what are the differences between affine and vector spaces ?

Please can you give me examples.

thanks

roger

Homework Helper
the essential difference is that approximately affine spaces have no origin. i think there are (at least) two ways to answer this, and i think you want this version.

http://mathworld.wolfram.com/AffineSpace.html

an example wolud be a straight line (not necessarily through the origin). if you like it is a vector subspace that has been shifted in some direction

Homework Helper
Exactly. In an "affine space" you have points and some kind of linear structure but no "zero" point and so can't add or subtract points as you can vectors.

For example, in R2, once we have set up a coordinate system, you can associate each point with the vector represented by an arrow from the origin to that point (exactly the kind of thing you do in Calculus). That gives you a "vector space". But that depends on the coordinates system-an there are an infinite number of different choices for a coordinate system. Without the coordinate system you just have R2 as an "affine space". You can calculate the distance between two points but you can't add or subtract points.