complementary subspace
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
$$U=V+W$$ 
if every $u\in U$ can be expressed as a sum
$$u=v+w$$ 
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
$$V\cap W=\{0\},$$ 
then we say that $V$ and $W$ form a direct sum^{} decomposition of $U$ and write
$$U=V\oplus W.$$ 
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 finitedimensional.
Proposition 1
Let $U\mathrm{,}V\mathrm{,}W$ be as above, and suppose that $U$ is finitedimensional. The subspaces $V$ and $W$ are complementary if and only if for every basis ${v}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{v}_{m}$ of $V$ and every basis ${w}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{w}_{n}$ of $W$, the combined list
$${v}_{1},\mathrm{\dots},{v}_{m},{w}_{1},\mathrm{\dots},{w}_{n}$$ 
is a basis of $U$.
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 $U$ are in a oneto 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 positivedefinite inner product^{}
$$\u27e8,\u27e9:U\times U\to \mathbb{K}.$$ 
In such circumstances, for every subspace $V\subset U$ we define the orthogonal complement^{} of $V$, denoted by ${V}^{\u27c2}$ to be the subspace
$${V}^{\u27c2}=\{u\in U:\u27e8v,u\u27e9=0,\text{for all}v\in V\}.$$ 
Proposition 2
Suppose that $U$ is finitedimensional and $V\mathrm{\subset}U$ a subspace. Then, $V$ and its orthogonal complement ${V}^{\mathrm{\u27c2}}$ determine a direct sum decomposition of $U$.
Note: the Proposition^{} is false if either the finitedimensionality or the positivedefiniteness assumptions^{} are violated.
Title  complementary subspace 

Canonical name  ComplementarySubspace 
Date of creation  20130322 12:52:16 
Last modified on  20130322 12:52:16 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  11 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 15A03 
Synonym  algebraic complement 
Defines  complementary 
Defines  direct sum 
Defines  decomposition 
Defines  orthogonal complement 