Linear and multilinear algebra; matrix theory Definition

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 $\R$ .

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Vector space definition
Linear subspace definition
Quotient space definition

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Tag	`0SVR29SV1V`
URI	`m2://0SVR29SV1V`
Kind	definition
Writing	W1 stated
MSC	15A03
Defines	Quotient space
Uses	Linear subspace

Quotient space

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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