Affine space description

topsquark

Forum Staff
Jan 2006
11,602
3,457
Wellsville, NY
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
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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.
 
  • Like
Reactions: 2 people

topsquark

Forum Staff
Jan 2006
11,602
3,457
Wellsville, NY
Thank you. That clears up a lot of things for me.

-Dan
 
Jul 2005
658
102
My house.
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

To further set things into perspective, affine spaces came to be by studying the tangent planes on a smooth manifold in R^n. These planes are not linear subspaces of R^n as they possibly do not cross the origin, but they are translations of subspaces of R^n.