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?
Best Answer
In fact, regardless of what others are saying, the two definitions of addition are equivalent for cosets of a subspace.
We need different notation for the two additions. Let's say $$(a+U)+(b+U)=(a+b)+U$$is the "coset-wise sum", while $$A\oplus B=\{a+b:a\in A, b\in B\}$$ is the "subset sum".
Proof: Suppose first that $x\in (a+U)\oplus (b+U)$. The definitions show that there exist $u_1,u_2\in U$ with $x=(a+u_1)+(b+u_2)$; hence $x=(a+b)+(u_1+u_2)\in (a+b)+U=(a+U)+(b+U)$.
Otoh if $x\in (a+U)+(b+U)=(a+b)+U$ then $x=(a+b)+u=(a+u)+(b+0)\in (a+U)\oplus (b+U)$.