Residual Classes and Quotient Spaces in Linear Algebra 📂Linear Algebra

Residual Classes and Quotient Spaces in Linear Algebra

Definition¹

Let’s refer to $V$ as $F$ -vector space, and $W \le V$ as subspace. For $v \in V$ , the following set

$\left\{ v \right\} + W := \left\{ v + w : w \in W \right\}$

is called the coset of $W$ containing $v$ . $+$ is the sum of sets.

Explanation

We often abbreviate $\left\{ v \right\} + W$ as $v + W$ .

Considering the set of all cosets of $W$ , denoted by $\left\{ v + W : v \in V \right\}$ , we define addition and scalar multiplication (by $F$ ) as follows:

$(v_{1} + W) + (v_{2} + W) = (v_{1} + v_{2}) + W,\quad \forall v_{1}, v_{2} \in V$

$a(v + W) = av + W\quad \forall v \in V \text{ and } a \in F$

Then, this set again forms a $F$ -vector space. This vector space is denoted by $V/W$ , and is called the quotient space of $V$ modulo $W$ .

Theorem

(a) $v + W$ being a subspace of $V$ is equivalent to $v \in W$ . (proof in algebra)

(b) For $v_{1}, v_{2} \in V$ , $v_{1} + W = v_{2} + W$ being true is equivalent to $v_{1} - v_{2} \in W$ . (proof in algebra)

(c) $V/W$ is a vector space, and the zero vector is $0_{V} + W = W$ . ( $0_{V}$ is the zero vector of $V$ .)

Proof

(a)

Assuming $(\Longrightarrow)$
Assume $v + W$ is a subspace of $V$ . Then, considering $0_{V}$ as the zero vector of $V$ , $0_{V} \in v + W$ holds. Therefore, for some $w \in W$ , $0_{V} = v + w$ and $w = -v \in W$ hold. $W$ being a subspace of $V$ is closed under scalar multiplication, so $v = -(-v) \in W$ holds.
Assuming $(\Longleftarrow)$
Assume $v \in W$ . To show $v + W$ is a subspace of $V$ , it suffices to show closure under addition and scalar multiplication. Let’s say $v + w_{1}, v + w_{2} \in v + W$ . Adding these two gives:
$(v + w_{1}) + (v_{1} + w_{2}) = v + (v + w_{1} + w_{2})$
Since $W$ is a subspace, it’s closed under addition, and by assumption $v$ is an element of $W$ , so for some $w_{3} \in W$ , the following holds:
$v + (v + w_{1} + w_{2}) = v + w_{3} \in W$
Now, let’s say $a \in F$ . Then similarly, by assumption, for some $w_{4} \in W$ , the following holds:
$a(v + w) = v + \left( (a-1)v + aw \right) = v + w_{4} \in W$

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(b)

Assuming $(\Longrightarrow)$
Assume $v_{1} + W = v_{2} + W$ . Then, for the zero vector $0_{V}$ of $V$ and some $w \in W$ , the following holds:
$v_{1} + 0_{V} = v_{2} + w \implies v_{1} - v_{2} = w \in W$
Assuming $(\Longleftarrow)$
Assume $v_{1} - v_{2} \in W$ . Then:
$\begin{align*} v_{2} + W &= \left\{ v_{2} + w : w \in W \right\} \\ &= \left\{ v_{2} + \left( (v_{1} - v_{2}) + w \right) : w \in W \right\} \\ &= \left\{ v_{1} + w : w \in W \right\} \\ &= v_{1} + W \end{align*}$

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