8.4.  Orthogonal decomposition

8.4.1 Definition:

Let S be any subset of V and let
             
It is easy to check that is the subspace of V, called the orthogonal complement of S..

 
8.4.2 Theorem (Orthogonal complement):

Let W be any subspace of V. Then Further, every vV can be uniquely written  as
w + v for some w
 W, u, we write this as  : V= W .
 

 Proof

8.4.3 Example:
(i)   Let V = with the standard inner-product. Let    
                  S = {( 1, 0, 0 ), ( 0, 1, 0 )}.      
      Then      
                  = {( x, y, z ) | (x, y, z) . (1, 0, 0 ) = 0 =  ( 0, 1, 0 ).( x, y, z )}   
                       =  {(0, 0, z ) | z }   
      If we write     
                  W = [S] = {( x, y, 0 ) | x, y }, then  = {( 0, 0, 3 ) | z}.       
        
                                         
(ii)   Let , the space of all real polynomials of degree at most 3, with inner product : 
                    
       Let  ,       
       Then  if and only if    
       This gives   
                      
        and 
                                         
        i.e.    
                        .          
         This system of equations in variables a, b, c, leads to solutions :      
                         
           where  are arbitrary. Hence  if and only if    
                          
           Hence   
                             
(iii)      Let    
           The vectors ( 2, 5, -1 ), ( -2, 1, 1 ) are orthogonal .   
           Then an orthonormal basis of W is given by        
                                     
           Then the projection  of any vector onto W is given by  
                           
            For example, if u = ( 1, 2, 3 ), then     
                         
              Thus      
                                                         
Let V be an inner-product space and let W be a subspace of V.  Theorem 8.1.4 tells us that  , i.e. , the space V decomposes as a sum of two orthogonal  subspaces. Decompositions, as given by theorem 8.4.2 are very useful in understanding linear transformation on inner-product spaces ( see proposition 9.3.2 and theorem 10.2.4. ). This idea can be extended as follows :     
 
8.4.4 Definition:
Let V be an inner-product space and be subspaces of  V such that       
          
   In that case, we say V is an orthogonal sum of the subspaces and we write  it as     
             
   The following is easy to prove :                
 
8.4.5 Proposition:

Let Then there exist unique such that  
  

 Proof

 
As another application of the projections, we have the following :
 
8.4.6 Theorem (Best unique approximation):

Let W be any subspace of V and let  vV. Then there exists a unique such that
       

 Proof

 
8.4.7 Example:
(i)   Let V = and W be the subspace of generated by {( 2, -1, -2 ), ( 1, 0, 1 )}.     
       Then, the best approximator for u = ( 2, 1, 3 ) from W is given by    
            
       The distance of  u =( 2, 1, 3 ) from W is given by       
           
(ii)   Consider    the space of all real polynomials of degree at most 3, with inner-product
        
       Let W  denote the subspace of V  spanned by { 1, t }. Since         
        , { 1, t } forms an orthogonal basis of  w .            
       Since      
         the orthogonal projection onto W ,         

        the best approximator , is given by     
              

 
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