Geometry - Affine Spaces

Vector Spaces, aka Linear Algebra.

We have some things that look like Descartes' points, though we don't know exactly what they are.

They are linear and we call them vectors.

• v =1 a1 v1 +1 a2 v2 is a vector

Generators

• How few vectors can I use to create all of them?
• The number is the dimension.
• A set having that number, with appropriate weasel words, is a basis.
• With respect to a basis (v1, v2, ...) any vector has coordinates (a1, a2, ... ), which we write v =1 a1 v1 +1 a2 v2 +1 ...

Transformations

Vectors can be transformed,

• Rigid transformations in 2D
1. Unit
2. Rotation
3. Reflection

More complex in 3D

• Non-rigid transformations in 2D
1. Scale
2. Skew

These transformations are all linear.

• They can be represented as matrices. That is v' =1 Tv means the same as (a1', a2', ... ) = (a1, a2, ... ) M.
• The transformations can equally well operate on the basis v' =1 Tv =1 a1 Tv1 +1 a2 Tv2 +1 ...

If you are Felix Klein you show that they are a group, and how they separate into subgroups. But even though you are not Felix Klein your response to this formulation is the same.

Affine Spaces

Intuition is that when I add an origin, O, vectors define points.

• Intuitively we can identify each point, P, with the vector that joins it to the origin.
• Therefore we add one point O, and get as many points as there are vectors
• P =2 O +2 v