Linear and multilinear algebra; matrix theory Theorem

Second isomorphism theorem for vector spaces

Builds on Quotient space Sum of subspaces Intersection of subspaces

Statement

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $U$ and $W$ be subspaces of $V$ . Then the map $U / (U \cap W) \to (U + W) / W, \qquad u + (U \cap W) \mapsto u + W,$ is well defined, and it is a linear isomorphism. Here $U + W$ is the sum of the subspaces, and both sides are quotient spaces.

Remarks

In finite dimensions, it gives Grassmann's formula: $\dim U - \dim(U \cap W) = \dim(U + W) - \dim W$ . The proof applies the first isomorphism theorem to the map $U \to (U + W)/W$ , $u \mapsto u + W$ , which is surjective with kernel $U \cap W$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Vector space definition
Linear subspace definition
Intersection of subspaces statement
Quotient space definition
Sum of subspaces definition
Second isomorphism theorem for vector spaces statement

Tag	`0APN65D6Y6`
URI	`m2://0APN65D6Y6`
Kind	statement, theorem
Writing	W0 stub
MSC	15A03, 16D10
Status	This copy of the site has no build data. Run `m2 status 0APN65D6Y6`.
Uses	Quotient space, Sum of subspaces, Intersection of subspaces

Second isomorphism theorem for vector spaces

Statement

Remarks

Reading path

Further reading

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