Affine Subsets

Overview

Let $V$ be a vector space and $U\le V$. The left coset of $U$ containing $v\in V$ is

An affine subset of $V$ is a subset of $V$ of form $v+U$. Such a subset is said to be parallel to $U$. Since affine subsets correspond to cells of an equivalence relation $\sim$, any two affine subsets of $U$ are either parallel or disjoint. In particular, $v\sim w$ if and only if $v-w\in U$.

Quotient Spaces

Let $U$ be a vector subspace of $V$. Then the quotient space $V/U$ is the set of all affine subsets of $V$ parallel to $U$. That is,

This set is a vector space under the following definitions of addition and scalar multiplication:

Quotient Map

Let $U$ be a vector subspace of $V$. The quotient map $\pi$ is the linear map $\pi :V\to V/U$ given by

Let $T\in \mathcal{L}(V,W)$. The quotient map $\pi :T\to T/(\text{null T})$ corresponds to the canonical homomorphism of $T$.