Page 86 - 35Linear Algebra

86 Vectors in Space, n-Vectors

The plane determined by two vectors u and v can be written as

{P + su + tv | s, t ∈ R} .

Example 46 (A plane in a higher dimensional space)

  
   
 3 1 0  
  
 1     
0
1


 

      
0
4
0
 
     
  + s   + t   s, t ∈ R
0
 1    
0
 
      

 5     
0
0
 
 

  
9 0 0 
 
describes a plane in 6-dimensional space parallel to the xy-plane.
Parametric Notation
We can generalize the notion of a plane with the following recursive def-
inition. (That is, infinitely many things are defined in the following line.)
n
Definition A set of k + 1 vectors P, v 1 , . . . , v k in R with k ≤ n determines
a k-dimensional hyperplane,
k
( )
X
P + λ i v i | λ i ∈ R
i=1
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