Page 88 - 35Linear Algebra

88 Vectors in Space, n-Vectors

You might sometimes encounter the word “hyperplane” without the qual-
ifier “k-dimensional. When the dimension k is not specified, one usually as-
n
sumes that k = n − 1 for a hyperplane inside R . This is the kind of object
that is specified by one algebraic equation in n variables.

Example 48 (Specifying a plane with one linear algebraic equation.)
The solution set to

   
x 1 1 − x 2 − x 3 − x 4 − x 5
x 2 x 2
   
   
x 1 + x 2 + x 3 + x 4 + x 5 = 1 ⇔  x 3  =  x 3 
   
x 4 x 4
   
x 5 x 5
is

 







 
 1 −1 −1 −1 −1 
  

 0  1   0   0   1  


 
          
0 0 1 0 0 ,
  + s 2   + s 3   + s 4   + s 5   s 2 , s 3 , s 4 , s 5 ∈ R
         
 0  0   0   1   0  

 


  
0 0 0 0 1 
 
5
a 4-dimensional hyperplane in R .
4.3 Directions and Magnitudes
Consider the Euclidean length of an n-vector:
v
n
u
p uX
n 2
i 2
1 2
kvk := (v ) + (v ) + · · · + (v ) = t (v ) .
2 2
i=1
Using the Law of Cosines, we can then figure out the angle between two
n
vectors. Given two vectors v and u that span a plane in R , we can then
connect the ends of v and u with the vector v − u.
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