quotient of a vector space

[[concept]]

Quotient

Let $W \subseteq V$ be a subspace. We define an equivalence relation on $V$ by

v \sim v^{'} ⟺ v - v^{'} \in W

Define $[v] = {v^{'} \in V : v^{'} \sim v}$ the equivalence class of $v$ . Then

V | W = {[v] : v \in V}

is called the quotient space which we typically call " $V$ mod $W$ " and notate as

[v] = v + W

$V | W$ is a vector space such that for all $v_{1}, v_{2} \in V$ and $λ \in K$

$(v_{1} + W) + (v_{2} + W) := (v_{1} + v_{2}) + W$
$λ (v + W) = λ v + W$

Note

$W = 0 + W = w + W \forall w \in W$

Mentions

File	Last Modified
Functional Analysis Lecture 3	2025-10-02

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}