The quotient vector space $V\mathrm{/}W$ and its canonical projection

Let $V$ be a vector space over a field $F$ and let $W$ be a linear subspace of $V$ (Vector space over a field, Linear subspace of a vector space). For $v\in V$, the coset of $W$ represented by $v$ is $v+W:=\{v+w:w\in W\}\mathrm{.}$ The quotient set is $V\mathrm{/}W:=\{v+W:v\in V\}$. Its addition and scalar multiplication are $(v+W)+(u+W):=(v+u)+W,a(v+W):=(av)+W\mathrm{.}$ The resulting vector space is the quotient vector space of $V$ by $W$. The canonical projection is $\pi :V⟶V\mathrm{/}W,\pi (v):=v+W\mathrm{.}$ The independence of the displayed operations from their representatives, the vector-space axioms, and the linearity and kernel of $\pi$ are established in Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ ↗.

Quotient spaces enter this development because the reverse triangularisation argument descends an operator from $V$ to the quotient by an invariant eigenline $⟨v⟩$. The quotient removes that line while retaining the induced linear action needed for induction.