quotient of a vector space

Let $W\subseteq V$ be a subspace. We define an equivalence relation on $V$ by

Define $[v]=\{v^{\prime }\in V:v^{\prime }\sim v\}$ the equivalence class of $v$. Then

is called the quotient space which we typically call "$V$ mod $W$" and notate as

$V|W$ is a vector space such that for all $v_1,v_2\in V$ and $\lambda \in \mathbb{K}$

1. $(v_1+W)+(v_2+W):=(v_1+v_2)+W$
2. $\lambda (v+W)=\lambda v+W$

$$W=0+W=w+W\mathrm{\forall }w\in W$$