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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 $U,V,W$ be as above, and suppose that $U$ is finitedimensional. The subspaces $V$ and $W$ are complementary if and only if for every basis $v_{1},\ldots,v_{m}$ of $V$ and every basis $w_{1},\ldots,w_{n}$ of $W$, the combined list
$v_{1},\ldots,v_{m},w_{1},\ldots,w_{n}$ 
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
$\langle,\rangle:U\times U\rightarrow\mathbb{K}.$ 
In such circumstances, for every subspace $V\subset U$ we define the orthogonal complement of $V$, denoted by $V^{\perp}$ to be the subspace
$V^{\perp}=\{u\in U:\langle v,u\rangle=0,\text{ for all }v\in V\}.$ 
Proposition 2.
Suppose that $U$ is finitedimensional 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 finitedimensionality or the positivedefiniteness assumptions are violated.
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Comments
Adding to Your Entry
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
Complementary spaces are unique?
Regarding the remark following Proposition1: I don’t believe that, given $U$, subspace^{} of $V$, that $U$ has a unique complement in $V$. E.g. any two lines in $\mathbb{R}^{2}$ are complementary^{}. Did I misunderstand something here?