TITLE: Why is addition defined, and not implied, on quotient spaces?
QUESTION [16 upvotes]: Small question. In chapter 3, section E, page 96 of "Linear Algebra Done Right", addition in quotient vector spaces is defined this way:

I understand why scalar multiplication has to be defined, because multiplying a subset of a vector space with a scalar was not defined. But why can addition of affine subsets be said to work this way, if the sum of subsets of a vector space was already defined?

$v+U, w+U$ are both subsets of a vector space $V$, so $(v+U)+(w+U)$ is the set containing all possible sums of elements of $v+U$ with elements of $w+U$. Shouldn't the statement $(v+U)+(w+U)=(v+w)+U$ be a theorem?

REPLY [3 votes]: Other answers give several good reasons already; here is one more, not yet mentioned.
Conceptually, we often think of the cosets $v+U$ (or more generally, equivalence classes in a quotient) as a different kind of thing from arbitrary subsets of a space.  We’re thinking of the coset/quotient space $V/U$ as a separate object in its own right, not as living inside the power set $\mathcal{P}(V)$, even though formally $V/U \subseteq \mathcal{P}(V)$.  So we think of the operations on $V/U$ as defined in themself — we don’t stop to consider whether they’re a special case of the operations on $\mathcal{P}(V)$.  In this case the two definitions do agree; but there are other places where the operations you use on a quotient don’t agree with the setwise operations on the power-set (e.g. cosets in a general group/ring/module, such as mod-$n$ arithmetic $\mathbb{Z}/n\mathbb{Z}$).
This way of thinking — viewing different sets as separate objects, even though they may have some elements in common — is widespread and important in maths, but often not explicitly discussed or taught. It’s more explicitly discussed in programming, as a kind of abstraction, or implementation-independence.  Formally, unicode-strings (or whatever) are just certain sequences of bytes.  But (in many modern programming languages) you declare them a separate type of object, and you deliberately don’t allow byte-sequence functions to act directly on unicode-strings.