FCM-Home       Orthogonal Splittings

   In this section we learn that every subspace    of a vector space    has an orthogonal complement  . Accordingly, every vector  x  in    can be "split up" uniquely as  x x + x, where  x  belongs to    and  x  belongs to  ; a fact which motivates the terminology  "  and    form a splitting of  ".

Here we establish the fundamental properties of orthogonal vector space splittings. Our key tools for this purpose are orthonormal bases and orthogonal projection operators. We begin by introducing the Gram-Schmidt orthonormalization process. It turns an arbitrary basis of a vector space into an orthonormal basis.

 
Theorem   (Gram-Schmidt Orthonormalization Process)   Given a linearly independent subset  {a, ... ,a}  of  , the set of vectors  {v, ... ,v}  defined below is an ONB of  span().       

Moreover,  span{a, ... ,a} span{v, ... ,v}  for each  1 .

Aided by the Gram-Schmidt process, we see that every nonzero subvector space    of    has an orthonormal basis and, more generally, that every orthonormal subset of    can be  complemented to an orthornormal basis  of  .

   Turning to orthogonal splittings of subspaces, say  , of  , we begin with some frequently used properties of the orthogonal complement operation.

Proposition   For a subspace    of  , the following hold:
  1. If  , then  < .   
  2. If    is a subset of  , then  span().    
  3. If  < <   are subspaces, then  {0}.    

A useful tool when working with orthogonal splittings is the projection operator defined below. 

Definition   Given subspaces  {0} <   of  , the orthogonal projection  of    onto    is  proj ,

proj(v) (v b)b + ... + (v b)b,

where  {b, ... ,b}  is an arbitrary ONB of  .
Proposition  Given subspaces  <   of  , every  v  in    has a unique decomposition

v v + v

where  v  is the component of  v  in  , and  v v proj(v)  is the component of  v  which is orthogonal to  .    

Orthogonal splittings of vectors are useful in many contexts. For example, they provide a convenient tool to conceptualize the Gram-Schmidt orthonormalization process.

Corollary  Given subspaces  <   of  ,
 
dim() dim() + dim().   

As a further consequence of the above proposition we see that, for a subset    of a subspace    of  , span().

Exercises   on orthonormal basis, orthogonal projections and orthogonal splittings