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complementary subspace (Definition)

Direct sum decomposition.

Let $ U$ be a vector space, and $ V,W\subset U$ subspaces. We say that $ V$ and $ W$ span $ U$, 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 $ V$ and $ W$ form a direct sum decomposition of $ U$ and write
$\displaystyle U=V\oplus W.$
In such circumstances, we also say that $ V$ and $ W$ are complementary subspaces, and also say that $ W$ is an algebraic complement of $ V$.

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

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

Let us also remark that direct sum decompositions of a vector space $ U$ are in a one-to correspondence fashion with projections on $ U$.

Orthogonal decomposition.

Specializing somewhat, suppose that the ground field $ \mathbb{K}$ is either the real or complex numbers, and that $ U$ is either an inner product space or a unitary space, i.e. $ U$ 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 $ V$, 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 $ U$ is finite-dimensional and $ V\subset U$ a subspace. Then, $ V$ and its orthogonal complement $ V^\perp$ determine a direct sum decomposition of $ U$.

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



"complementary subspace" is owned by rmilson.
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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, basis, finite-dimensional, characterization, addition, sum, span, subspaces, vector space
There are 108 references to this entry.

This is version 7 of complementary subspace, born on 2002-07-26, modified 2007-09-15.
Object id is 3209, canonical name is Complimentary.
Accessed 22412 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
Pending Errata and Addenda
None.
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Adding to Your Entry by pjohara15 on 2008-01-19 19:25:26
Hello -

I am wondering if you would mind my adding a short exposition to your post concerning Sum and Direct Sum.

I have proved some fundamental results and would like to post them with your addition and perhaps get some input.

Patrick J. O'Hara
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