Quotient Spaces

Definition(s).

A translate of a subspace $X$ by $v\in V$, denoted $v+X$, is defined as follows:

$$v+X:=\{v+x\mid x\in X\}.$$

The set of all translates $v+X$ forms a vector space called the quotient space and is denoted by $V/X$.

The canonical quotient map is defined as follows:

$$\pi :V\to V/X,v\mapsto v+X.$$

Remark.

If we consider the equivalence relation on $V$ such that $v\sim v^{\prime }$ if and only if $v-v^{\prime }\in X$, then $v+X$ is the equivalence class of $v$.

With the observation that $\pi$ is surjective, rank nullity shows that $dim(V)=dim(X)+dim(V/X)$, i.e. $dim(V/X)=dim(V)-dim(X)$. Thus $V/X$ has the same dimension as a complement of $X$. Equivalently, we can say that $V/X$ is isomorphic to any complement of $X$. It’s worth noting that there is no choice involved in $V/X$. We don’t need to choose any bases to span the space. Thus, the quotient space “behaves like a choice-free complement,” as my professor liked to say.

Quotient spaces are a great alternative way to think about the rank nullity theorem. The key idea is that for any $T\in \mathcal{L}(V,W)$, $V/null(T)$ is isomorphic to $range(T)$. Then, directly showing that $dim(V/X)=dim(V)-dim(X)$ is enough to show rank nullity. Since complements were used in our original proof for rank nullity, quotient spaces really do present themselves as a choice-free complement-alternative.