Page 87 - 35Linear Algebra

4.2 Hyperplanes 87

unless any of the vectors v j lives in the (k − 1)-dimensional hyperplane de-
termined by the other k − 1 vectors

( )
k
X
0 + λ i v i | λ i ∈ R .
i6=j

Example 47 (3+1 vectors that do not specify a 3-dimensional hyperplane)

        
 3 1 0 1  
  
1
1
 1       
0


 

        
0
0
4
0
 
       
S :=   + s   + t   + u   s, t, u ∈ R
0
0
 1      
0
 

        

 5       

0
0
0
 

  
9 0 0 0 
 
is not a 3-dimensional hyperplane because
  
        
1 1 0 1
 0  
  
1 0 1  0 1 
           
 
           
0 0 0  0 0 
         
  = 1   + 1   ∈ s   + t   s, t ∈ R .
0
0
0
0
0
         
 
           
0
0
0
0
       0   
  



  

0 0 0 0 0 

In fact, the set could be rewritten as
      
 3 1 0 
  

 1     
1
0

 

      

0
0
4
 
     
S =   + (s + u)   + (t + u)   s, t, u ∈ R
0
0
 1     
 
      
 5     
0
0


 

  
9 0 0 
 
      
 3 1 0 
  

 1     
0
1


 

      
0
4
0
 
     
=   + a   + b   a, b ∈ R
0
0
 1    
 
      

0

 5     

0
 

  
 
9 0 0 
6
and so is actually the same 2-dimensional hyperplane in R as in example 46.
87