A fundamental confusion between Affine space and quotient space

Question

I am reading the text book "Linear Algebra a geometric approach "by S Kumaresan. It says that a non-empty subset $S $ of Vector space $V $ is an affine space iff it is of the form $v+W $ for some $v \in V $ and a vector subspace $W $ of $V$. Now my confusion is that the definition of Quotient space says exactly same thing! Does they are same thing if yes why they are distinguished by distinct names. We know affine space my not a subspace but still $dim ~v+W =dim W$. How does one can define dimension of nonvector space? Thanks for reading.

Answer 1

No they are different concepts.

In the case of affine spaces by the notation $v + W$ it's meant the set $$ v + W := \left\{ v + w \mid w \in W \right\} $$ This set (affine space), that is subset of our original vector space in general will might not be vector space at all.

Now the quotient space over $W$ is the space of all those cosets/affine spaces.

$$ V/W := \left\{ v + W \mid v \in V \right\} $$

Note that now the elements of this space are sets of elements of the original space (or equivalence classes of the relation $ u \sim v \iff u - v \in W$).

The quotient space is a vector space with the scalar multiplication and addition defined as $$ (u + W) + (v + W) = u + v +W $$ $$ \lambda(u+W) = \lambda u + W $$ It's easy to show that those operations are well defined (they send equal equivalence classes into the same value).