Linear and multilinear algebra; matrix theory Theorem

Third isomorphism theorem for vector spaces

Statement

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $W \subseteq U$ be subspaces of $V$ . Then $U / W$ is a subspace of $V / W$ , and the map $(V / W) / (U / W) \to V / U, \qquad (v + W) + U/W \mapsto v + U,$ is well defined, and it is a linear isomorphism of quotient spaces.

Remarks

The proof applies the first isomorphism theorem to the surjective map $V / W \to V / U$ , $v + W \mapsto v + U$ , whose kernel is $U / W$ . In finite dimensions, both sides have dimension $\dim V - \dim U$ .

Reading path

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Vector space definition
Linear subspace definition
Quotient space definition
Third isomorphism theorem for vector spaces statement

Tag	`07RB3EMBWW`
URI	`m2://07RB3EMBWW`
Kind	statement, theorem
Writing	W0 stub
MSC	15A03, 16D10
Status	This copy of the site has no build data. Run `m2 status 07RB3EMBWW`.
Uses	Quotient space

Third isomorphism theorem for vector spaces

Statement

Remarks

Reading path

Further reading

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