This is almost the same question as Origin in vector space?, and I am also confused by the statements in the wikipedia article of Affine space:
St1: Roughly, affine spaces are vector spaces whose origin is not specified.
St2: When considered as a point, the zero vector is called the origin.
The St2 is used as an answer in some replies to the post Origin in vector space?. However, I am still confused.
My question is:
which one is the affine space really don't care, the origin (i.e. the zero vector) or the Origin point (a specific fixed point in the space, to build coordinates)?
If the origin just means the zero vector, and affine spaces means a space does not need zero vector (the unit of vector space), it is clear and acceptable by definition. But in the wikipedia article of Affine space (or other places introducing Affine space), we always mention Affine combination, which is independent of the choice of the Origin point. Here the Origin point means a specific fixed point in the space, which is not a vector (One may argue it can be treated as zero vector stick to this point, but a vector does not need to stick to any point). If it means the origin (i.e. the zero vector), then how does it connect to the Affine combination (which is independent of the choice of the Original point)
If it means Origin point, it does not make sense. Because the definition of vector space itself does not need a fixed reference origin point. In fact, an vector is configured by the length and the direction, we don't really need an origin point for any vector. A vector can be originated from any place, not necessarily from a specific "Origin". (Sometimes we may need the help of a fixed Origin point when we use vector as the position vectors of all the other points, but the vectors themselves do not need the Origin point).