Page 91 - 35Linear Algebra

4.3 Directions and Magnitudes 91

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Example 52 of vectors from R that are orthogonal to each other.
   
1 1
1
   −1 
   
1

   1  = 0.
  · 
 .  .
. .
 .   . 
1 −1
n
n
Notice that the zero vector 0 n from R is orthogonal to every vector in R ;
n
0 n · v = 0 for all v ∈ R .
The dot product has some important properties; it is
1. symmetric:
u v = v u ,

2. Distributive:
u (v + w) = u v + u w ,

3. Bilinear (which is to say, linear in both u and v):

u (cv + dw) = c u v + d u w ,

and
(cu + dw) v = c u v + d w v .

4. Positive Definite:
u u ≥ 0 ,
and u u = 0 only when u itself is the 0-vector.

There are, in fact, many different useful ways to define lengths of vectors.
Notice in the definition above that we first defined the dot product, and then
defined everything else in terms of the dot product. So if we change our idea
of the dot product, we change our notion of length and angle as well. The
dot product determines the Euclidean length and angle between two vectors.
Other definitions of length and angle arise from inner products, which
have all of the properties listed above (except that in some contexts the
positive definite requirement is relaxed). Instead of writing for other inner
products, we usually write hu, vi to avoid confusion.

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