# The canonical projection uniquely determines a vector space structure on the quotient space

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.

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I have the vector space structure, I am just having a hard time proving uniqueness – user61466 Feb 8 '13 at 10:17
OK - you should ideally say something about this in the body of the question. – Matthew Pressland Feb 8 '13 at 10:18
Sorry about that. This is my first post here. So the structure is just the standard addition and scalar multiplication on the element $v$ – user61466 Feb 8 '13 at 10:21
No worries. I'm not sure what you mean by standard addition and scalar multiplication on the element $v$. Is $v$ even an element of $V/W$? – Matthew Pressland Feb 8 '13 at 10:28
Addition: $(v+W)+(v'+W)= v+v'+W$,and scalar multiplication is $c(v+W)=cv+W$ – user61466 Feb 8 '13 at 10:31

Rather than pick two vector space structures such that $\pi$ is linear with respect to both, and then showing they must be the same, it's a bit clearer simply to show that $\pi$ forces our hand in the definition.
For example, for the addition, we try to define $(v+W)+(v'+W)$. But this is $\pi(v)+\pi(v')$, and we're insisting that our definition makes $\pi$ linear, so this has to be equal to $\pi(v+v')$, which is $(v+v')+W$. There's simply nothing else we can do. You should be able to construct a similar argument for the scalar multiplication.
If you're more comfortable showing that any two vector space structures on $V/W$ such that $\pi$ is linear coincide, you can also modify this argument to say it that way.