complementary subspace (original) (raw)

Direct sum decomposition.

Let U be a vector space, and V,W⊂U subspaces. We say thatV and W span U, and write

if every u∈U can be expressed as a sum

for some v∈V and w∈W.

If in addition, such a decomposition is unique for all u∈U, or equivalently if

then we say that V and W form a direct sum decomposition of Uand write

In such circumstances, we also say that V and Ware complementary subspaces, and also say that W is an algebraic complement of V.

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 v1,…,vm of V and every basisw1,…,wn of W, the combined list

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.

Proposition 2

Suppose that U is finite-dimensional and V⊂U a subspace. Then, V and its orthogonal complement V⟂ determine a direct sum decomposition of U.