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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

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    $\begingroup$ 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. $\endgroup$
    – Mark
    Jun 7, 2021 at 12:52

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They are the same.

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 $v'\sim v$.

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