The orthogonal decomposition of a vector in is the sum of a vector in a subspace of and a vector in the orthogonal complement to .
The orthogonal decomposition theorem states that if is a subspace of , then each vector in can be written uniquely in the form
where is in and is in . In fact, if is any orthogonal basis of , then
and .
Geometrically, is the orthogonal projection of onto the subspace and is a vector orthogonal to