Coset and Quotient Space

    이 포스트에서 $V, W$는 모두 $F$-벡터공간으로 취급한다.

Coset

Definition 1. Let $W \leq V$. $\forall v \in V$, the set $\{v\} + W := \{v + w \,|\, w \in W\}$ is called the coset of $W$ containing $v$. 
It is customary to denote this coset by $v + W$ rather than $\{v\} + W$.

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Theorem 1

Theorem 1. Let $W \leq V$, and let $v + W$ be a coset of $W$ containing $v$. 
(a) $v + W \leq V \Longleftrightarrow v \in W.$
(b) Let $v_1, v_2 \in V.$ Then $v_1 + W = v_2 + W \Longleftrightarrow v_1 - v_2 \in W.$

Proof. 
(a) Assume that $v + W$ is a subspace of $V$. Let $u \in v + W$. Then $\exists w \in W$ such that $u = v + w$. Note that $u + v = v + (v + w) \in v + W.$ Since $W$ is a subspace of $V$, $v \in W$.
Assume that $v \in W$. Then $v + (-v) = \mathbf{0} \in v + W$. Let $x, y \in v + W$ and $c \in F$. Then $\exists w_1, w_2 \in W$ such that $x = v + w_1, y = v + w_2.$ Hence $cx + y = c(v + w_1) + (v + w_2) = v + (cv + cw_1 + w_2).$ Since $cv + cw_1 + w_2 \in W$, $cx + y \in v + W$. Thus $v + W$ is a subspace of $V$. 

(b) Assume that $v_1 + W = v_2 + W$. Let $u \in v_1 + W$. Then $\exists w_1 \in W$ such that $u = v_1 + w_1$. Since $u \in v_2 + W$, $\exists w_2 \in W$ such that $u = v_2 + w_2$, and so $u = v_1 + w_1 = v_2 + w_2.$ Hence $(v_1 - v_2) + (w_1 - w_2) = \mathbf{0} \in W.$ Since $(w_1 - w_2) \in W$, $(v_1 - v_2) \in W$. 
Assume that $v_1 - v_2 \in W$. Let $u \in v_1 + W$. Then $\exists w \in W$ such that $u = v_1 + w$. Note that $u = v_2 + (v_1 - v_2 + w) \in v_2 + W$. Thus $v_1 + W \subseteq v_2 + W.$ In the same manner, it is easily seen to $v_2 + W \subseteq v_1 + W$. Thus $v_1 + W = v_2 + W.$ $\blacksquare$

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Quotient Space

Definition 2. We define the quotient space of $V$ modulo $W$, denoted $V \backslash W$, by $V \backslash W = \{v + W \, | \, v \in V\}$. 

Theorem 2

Theorem 2. The quotient space of $V$ modulo $W$ is a vector space with the following operations: $$(v_1 + W) + (v_2 + W) = (v_1 + v_2) + W, \forall v_1, v_2 \in V \\ a(v + W) = av + W, \forall v \in V, a \in F.$$

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Reference is here: https://product.kyobobook.co.kr/detail/S000003155051

Linear Algebra | Stephen Friedberg - 교보문고