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

$$U=V+W$$

ifevery $u\in U$ 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 $u\in U$, orequivalently 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 $U$ isfinite-dimensional.

Proposition 1

Let $U\mathrm{,}V\mathrm{,}W$ be as above, and suppose that $U$ isfinite-dimensional. The subspaces $V$ and $W$ are complementary ifand only if for every basis $v_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{v}_m$ of $V$ andevery 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 eitherthe real or complex numbers, and that $U$ is either an inner productspace or a unitary space, i.e. $U$ comes equipped with apositive-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 orthogonalcomplement $V^{\mathrm{⟂}}$ determine a direct sum decomposition of $U$.

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