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 finite-dimensional.

Proposition 1

Let $U\mathrm{,}V\mathrm{,}W$ be as above, and suppose that $U$ is finite-dimensional. The subspaces $V$ and $W$ are complementary if and only if for every basis $v_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{v}_m$ 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 one-to correspondence fashion with projections on $U$.

Orthogonal decomposition.

Specializing somewhat, suppose that the ground field $𝕂$ 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

$$⟨,⟩:U\times U\to 𝕂.$$

In such circumstances, for every subspace $V\subset U$ we define the orthogonal complement of $V$, denoted by $V^{⟂}$ to be the subspace

$$V^{⟂}=\{u\in U:⟨v,u⟩=0,\text{ for all }v\in V\}.$$

Proposition 2

Suppose that $U$ is finite-dimensional and $V\mathrm{\subset }U$ a subspace. Then, $V$ and its orthogonal complement $V^{\mathrm{⟂}}$ determine a direct sum decomposition of $U$.

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

Title	complementary subspace
Canonical name	ComplementarySubspace
Date of creation	2013-03-22 12:52:16
Last modified on	2013-03-22 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