# Definition of quotient space: equivalence classes vs affine subsets

Suppose $$V$$ is a vector space over any field and $$U$$ its subspace. Consider two different definitions of the quotient space $$V/U$$:

1. $$V/U = \{v + U \mid v \in V\}$$, i.e. the set of all affine subsets of $$V$$ parallel to $$U$$. This definition is used in Axler's Linear Algebra Done Right 3rd edition section 3.E.
2. Define an equivalence relation $$\sim$$ on $$V$$: $$v \sim v' \text{ iff } v - v' \in U$$. Define $$V/U$$ to be the set of all equivalence classes generated by $$\sim$$. This definition is used in Hackbusch's Tensor Spaces and Numerical Tensor Calculus 3.1.3.

What advantages and disadvantages do these two definitions have compared to each other

• They are not really different definitions. Note that $v+U=w+U$ happens if and only if $v-w\in U$. So the equivalence class of $v$ by the relation $\sim$ is exactly the set $v+U$. So the only difference is that in the first definition you just define the elements of $V/U$ directly (which gives a natural equivalence relation: $v\sim w$ iff $v+U=w+U$), in the second one you first define an equivalence relation. The only difference is what you define first: the equivalence relation or the equivalence classes.
– Mark
Jun 7, 2021 at 12:52

Whenever $$v'\sim v$$, then $$v'=v+v'-v\in v+U$$. Conversely, if $$v'\in v+U$$, then $$v'-v\in v-v+U=U$$, hence $$v'\sim v$$.