Let $V$ be a vector space and $W$ a subvector space of $V$. Prove that the canonical map $\pi: V \to V/W$ given by $\pi(v)=v+W$ such that $\pi$ is a linear map uniquely determines a vector space structure on $v/W$
I found the structure. It is just the standard addition and scalar multiplication on the element $v$ from the vector space $V$. Now I am just stuck on showing uniqueness.