Linear and multilinear algebra; matrix theory Theorem

Existence of a complementary subspace

Statement

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $U$ be a subspace of $V$ . Then $U$ has a complement in $V$ : there is a subspace $W$ of $V$ with $V = U \oplus W$ .

Remarks

Proof: extend a basis $B_U$ of $U$ to a basis $B$ of $V$ , and let $W$ be the span of $B \setminus B_U$ . For infinite-dimensional $V$ , this uses the extension of independent sets to bases, and hence Zorn's lemma. The statement fails for modules over rings: the submodule $2\Z$ of $\Z$ has no complement. For closed subspaces of topological vector spaces, a closed complement need not exist.

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
Sum of subspaces definition
Direct sum of subspaces definition
Complementary subspaces definition
Existence of a complementary subspace statement

Tag	`0P9T8WHNMH`
URI	`m2://0P9T8WHNMH`
Kind	statement, theorem
Writing	W0 stub
MSC	15A03
Status	This copy of the site has no build data. Run `m2 status 0P9T8WHNMH`.
Uses	Complementary subspaces

Existence of a complementary subspace

Statement

Remarks

Reading path

Further reading

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