A vector space projection onto a subspace 
of a vector space 
is a linear map 
satisfying 
for every 
. If 
is 
-dimensional and 
has inner product 
, an orthogonal projection can be defined as follows.
The most familiar projection is when 
is the x-axis in the plane.
In this case, 
is the projection. This projection is an orthogonal projection.
If the subspace 
has an orthonormal basis 
then
 |
is the orthogonal projection onto 
. Any vector 
can be written uniquely as 
, where 
and 
is in the orthogonal
subspace 
.
A projection is always a linear transformation and can be represented by a projection matrix. In addition, for any projection, there is an inner product for which it is an orthogonal projection.
