I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by Paul Bamberg and Shlomo Sternberg. In Chapter 1 authors define affine space and writes:
The space R2 is an example of a vector space. The distinction between vector space R2 and affine space AR2 lies in the fact that in R2 the point (0,0) has a special significance ( it is the additive identity) and the addition of two vectors in R2 makes sense. These do not hold for AR2.
How come AR2 has point (0,0) without special significance? and why the addition of two vectors in AR2 does not make sense? Please give concrete examples instead of abstract answers . I am a physics major and have done courses in Calculus, Linear Algebra and Complex Analysis.
Consider the vector space R3. Inside R3 we can choose two planes, P1 and P2. The plane P1 passes through the origin but the plane P2 does not. It is a standard homework exercise in linear algebra to show that the P1 is a sub-vector space of R3 but the plane P2 is not. However, the plane P2 resembles a 2-dimensional vector space in many ways, primarily in that it exhibits a linear structure. In fact, P2 is a classical example of an affine space.
One defect of the plane P2 is that it has no distinguished origin. One can artificially choose a point and redefine the algebraic operations in such a way to give it an origin, but that is not inherent to P2. Another problem is that the sum of two vectors in P2 is no longer in P2. One can think of AR2 as being modeled on this situation.