Quotient space

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a linear subspace of $V$. For $v \in V$, the coset of $v$ is $v + W = \{v + w : w \in W\}$. The quotient space $V / W$ is the set of all cosets $v + W$ with $v \in V$, with the operations $$(u + W) + (v + W) = (u + v) + W, \qquad a(v + W) = av + W$$ for $u, v \in V$ and $a \in F$.

Remarks

The operations are well defined, because $W$ is a subspace: the result does not depend on the choice of $u$ and $v$ in their cosets. Two cosets $u + W$ and $v + W$ are equal if and only if $u - v \in W$. The zero vector of $V / W$ is the coset $0 + W = W$. In $\R^2$ with $W$ the $x$-axis, the cosets are the horizontal lines, and $V / W$ is isomorphic to $\R$.

Explanations

L1 human

In the quotient space $V / W$, we treat two vectors as the same when they differ by a vector of $W$. Each element of $V / W$ is a copy of $W$ shifted by some vector. Dividing the plane by a line gives the set of all lines parallel to it, which behaves like a single line.