Page 272 - 35Linear Algebra

272 Orthonormal Bases and Complements

So the set
   1   1 
1
 2 
 3 
 
 −1   1   1 
,
,
   2  3
0 −1 1 
    
 
 3 
0 0 −1
 
⊥
is an orthogonal basis for L . Dividing each basis vector by its length yields
  √ 
 1   1  3
√ √
 
 6 √ 6 
 2 
  
− √  
 1   √ 1  3 
2   6  ,  √ 6  ,

  ,  2
 3 
 0  − √  
 6  6 
 √ 
 
 0 0 − 3 
2
⊥
and orthonormal basis for L . Moreover, we have
     
 c x



   
c
y
       
⊥
4
R = L ⊕ L =   c ∈ R ⊕   ∈ R x + y + z + w = 0 ,
4
z
 c     
 
   
c w
   
4
a decomposition of R into a line and its three dimensional orthogonal complement.
⊥ ⊥
Notice that for any subspace U, the subspace (U ) is just U again. As
such, ⊥ is an involution on the set of subspaces of a vector space. (An invo-
lution is any mathematical operation which performed twice does nothing.)
14.7 Review Problems
Reading Problems 1 , 2 , 3 , 4
Gram–Schmidt 5
Webwork:
Orthogonal eigenbasis 6, 7
Orthogonal complement 8

0
λ 1
1. Let D = .
0 λ 2
(a) Write D in terms of the vectors e 1 and e 2 , and their transposes.

a b
(b) Suppose P = is invertible. Show that D is similar to
c d

1 λ 1 ad − λ 2 bc −(λ 1 − λ 2 )ab
M = .
ad − bc (λ 1 − λ 2 )cd −λ 1 bc + λ 2 ad
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