Quotient Space

definitionvector-spaces

Definition

If $U$ a subspace of $V$, then the quotient space ${}^V{/}_U$ is the vector space of all translates of $U$:

$${}^V{/}_U=\left\{v+U:v\in V\right\}.$$

We can define addition and scalar multiplication on ${}^V{/}_U$ as,

$$(v+U)+(w+U)=(v+w)+U,$$

and,

$$\lambda (v+U)=(\lambda v)+U.$$

Examples