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In pure set theory level, to construct the quotient set we need:

$1)$ A set $X$

$2)$ A equivalence relation $\thicksim$

$3)$ With $2)$ and $3)$, we can form the pair $(X, \thicksim)$.

$4)$ With $(X, \thicksim)$, we can form a patition on the set $X$ and then stablish the equivalence classes on $X$.

$5)$ Finally, the set of all equivalence classes is the desired quotient set $Y := X/\thicksim$ (*).

Now, in linear algebra level, we denote the quotient vector space as:

$$\mathcal{Y} = \mathcal{X}/\mathcal{Z}$$

When I compare the notations $X/\thicksim$ and $\mathcal{X}/\mathcal{Z}$, I feel a bit confuse. So my question is: why in linear algebra we do not denote the quotient vector space simply as $\mathcal{Y} = \mathcal{X}/\thicksim$? In other words, since $\mathcal{Z} \subset \mathcal{X}$, is the $\thicksim \hspace{0.5mm}\subset X$ (the symbol $\thicksim$ denotes a subset)?

(*) I interpret the quotient set as: given the set $X$, its elements are all there "interacting" together. With a equivalence relation on $X$ we can divide, factor or quotient the elements of $X$ into classes (we "separate the grains"), then form a whole new set $Y$"

Prime Mover
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M.N.Raia
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    The equivalence relation determined by a subspace $\mathcal{Z} \subseteq \mathcal{X}$ has equivalence classes given by cosets of $\mathcal{Z}$ in $\mathcal{X}$: for any two vectors $\mathbf{x}_1, \mathbf{x}_2 \in \mathcal{X}$, we have $\mathbf{x}_1 \sim \mathbf{x}_2$ if and only if $\mathbf{x}_1 - \mathbf{x}_2 \in \mathcal{Z}$. The canonical map $\mathcal{X} \twoheadrightarrow \mathcal{X} / \mathcal{Z}$ collapses every translate (coset) of $\mathcal{Z}$ to a point. The notation $\mathcal{X} / \mathcal{Z}$ contains more information than just the fact that its a quotient, but also by what. – Sammy Black Jul 23 '22 at 01:35
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    We are just using the same notation to describe two different but related concepts. The notation $X / \sim$ refers to a quotient of a set by an equivalence relation; the notation $\mathcal{X} / \mathcal{Z}$ refers to the quotient of a space by a subspace. The latter is defined in terms of the former. It's very common in mathematics to "abuse notation" in this way. – Mark Saving Jul 23 '22 at 01:36

1 Answers1

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We use $Y=X/Z$ as the notation for the quotient space because the equivalence relation $\sim$, or strictly speaking $\sim_{X,Z}$, is completely known once you know the subspace $Z$. The definition of the relation is: for every $v,w\in X$, we say $v\sim w$ if and only if $v-w\in Z$. Then, you can easily verify that $\sim$ so defined is an equivalence relation on $X$, depending on $Z$.

So, the notation $X/Z$ quickly conveys what is meant. Contrast with $Y=X/\sim$. How would I know which relation $\sim$ refers to (without further elaboration)? Math is about communicating ideas. Now, you may want to use notation like $Y/\sim_{X,Z}$ to denote the quotient set by the above defined equivalence relation. This is fine if you're first learning the subject. For example, some teachers when introducing linear algebra write $0_V,0_W$ or $+_V,+_W$ to denote the zeros/addition operation on different vector spaces $V,W$ so as to distinguish the difference between them, but after a few lectures, they'll drop this extra notation and leave it up to the student to decipher the (almost always obvious) meaning from context. It's the same with quotient vector spaces: given a vector space $X$ and a subspace $Z\subset X$, the set $X/Z$ denotes the quotient set for the above relation, and you know that as long as you're working with vector spaces, the defining property of the equivalence relation is never messed around with. Hence, it quickly gets the meaning across without excessive notation. Also, the operations on $X/Z$ are the 'obvious ones' (well they're obvious once you work them out in detail atleast once).

An obligatory remark: in different contexts, we use the same notation to denote different things. For example, if you start with a topological space $X$, and a subset $A\subset X$, then the notation $X/A$ means something completely different. There are no vector spaces around, so it's obviously not the above meaning. What is meant is we let $\sim_A$ denote the equivalence relation generated by $a_1\sim_A a_2$, for $a_1,a_2\in A$ (or more explicitly, the intersection of all equivalence relations on $X$ containing $\{(a_1,a_2)\in X\times X\,:\, a_1,a_2\in A\}$); then $X/A:= X/\sim_A$. This equivalence relation has the following equivalence classes: for any point $x\in X\setminus A$, $[x]=\{x\}$ is a singleton, whereas for any $a\in A$, $[a]=A$. So, this 'collapses $A$ to a point'.

Now, one can consider the following situation: $X$ is a topological vector space (e.g $\Bbb{R}^n$) and $Z\subset X$ a vector subspace. Then, the symbol $X/Z$ has two potential meanings, the first as a quotient of vector spaces, the second as a quotient of topological spaces collapsing the subset to a single point. The intended meaning should always be decided based on context. Context is always important.

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