Vector space/Linear subspace/Residue class space/Fact/Proof

Proof

The existence and uniqueness of the group structure on $V/U$, and the homomorphism property of the projection, follow from fact. For the projection to be linear, the scalar multiplication must satisfy

$\lambda [v]=[\lambda v].$

We have to show that by this condition, a well-defined scalar multiplication on $V/U$ is given. Let $v^{\prime }=v+u$, with $u\in U$. Then we have

$\lambda v^{\prime }=\lambda (v+u)=\lambda v+\lambda u,$

and the is equivalent to $\lambda v$. The well-definedness of the scalar multiplication on $V/U$ and the surjectivity of the mapping imply that we have a vector space structure, and that the mapping is also compatible with the scalar multiplication.

To fact