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 
