Various other curiosities

Equation of a Line in an Affine Space defined by two point P and Q and t a real

set of point of the form (1-t) P + t Q

Equation of a Plane in an Affine Space defined by three not collinear point P, Q and R

set of point of the form (1-s) ((1-t) P + t Q) +s R

Vector and affine linear subspace are resp. vector and affine space

V : Non empty subset which is stable through the addition and the multiplication by a real

A : such as the set of the difference between any element of A is a linear vector subspace

Example

Vector subspace of R4

/ x
| y
| z
\ 0

Affine Subspace of R4 : standard affine 3 space

/ x                        / x1 - x2
| y              v1 - v2 = | y1 - y2   belongs to the previous 
| z                        | z1 - z2   vector subspace of R4
\ 1                        \ 0