PlanetMath: complementary subspace

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complementary subspace

(Definition)

Direct sum decomposition.

Let

be a vector space, and $V,W\subset U$ subspaces. We say that

and

span

, and write

$\displaystyle U=V+W$

if every $u\in U$ can be expressed as a sum

$\displaystyle u=v+w$

for some $v\in V$ and $w\in W$ .

If in addition, such a decomposition is unique for all $u\in U$ , or equivalently if

$\displaystyle V\cap W=\{ 0\},$

then we say that

and

form a direct sum decomposition of

and write

$\displaystyle U=V\oplus W.$

In such circumstances, we also say that

and

are complementary subspaces, and also say that

is an algebraic complement of

Here is useful characterization of complementary subspaces if is finite-dimensional.

Proposition 1 Let be as above, and suppose that is finite-dimensional. The subspaces and are complementary if and only if for every basis $v_1,\ldots, v_m$ of and every basis $w_1,\ldots,w_n$ of , the combined list

$\displaystyle v_1,\ldots,v_m,w_1,\ldots,w_n$

is a basis of .

Remarks.

Since every linearly independent subset of a vector space can be extended to a basis, every subspace has a complement, and the complement is necessarily unique.
Also, direct sum decompositions of a vector space are in a one-to correspondence fashion with projections on .

Orthogonal decomposition.

Specializing somewhat, suppose that the ground field $\mathbb{K}$ is either the real or complex numbers, and that

is either an inner product space or a unitary space, i.e.

comes equipped with a positive-definite inner product

$\displaystyle \langle,\rangle:U\times U\rightarrow \mathbb{K}.$

In such circumstances, for every subspace $V\subset U$ we define the orthogonal complement of

, denoted by $V^\perp$ to be the subspace

$\displaystyle V^\perp = \{ u\in U: \langle v,u\rangle = 0,$ for all $\displaystyle v\in V\}.$

Proposition 2 Suppose that is finite-dimensional and $V\subset U$ a subspace. Then, and its orthogonal complement $V^\perp$ determine a direct sum decomposition of .

Note: the Proposition is false if either the finite-dimensionality or the positive-definiteness assumptions are violated.

"complementary subspace" is owned by rmilson. [ full author list (2) ]

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Other names:

algebraic complement

Also defines:

complementary, direct sum, decomposition, orthogonal complement

Attachments:: theorem for the direct sum of finite dimensional vector spaces (Theorem) by matte
topological complement (Definition) by asteroid

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Cross-references: proposition, inner product, unitary space, inner product space, complex numbers, real, ground field, projections, complement, subset, linearly independent, basis, finite-dimensional, characterization, addition, sum, span, subspaces, vector space
There are 110 references to this entry.

This is version 8 of complementary subspace, born on 2002-07-26, modified 2008-06-01.
Object id is 3209, canonical name is Complimentary.
Accessed 23847 times total.

Classification:

AMS MSC:

15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

Pending Errata and Addenda

None.[ View all 1 ]

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