Page 307 - Matrix Analysis & Applied Linear Algebra

5.4 Orthogonal Vectors 303
Exercises for section 5.4

5.4.1. Using the standard inner product, determine which of the following pairs
are orthogonal vectors in the indicated space.
   
1 −2
3
(a) x =  −3  and y =  2  in ,
4 2
i 0
   
4
 1+i 
 1+i 
(b) x =   and y =   in C ,
2 −2
1 − i 1 − i
1 4
   
4
 −2 
(c) x =   and y =   in ,
 2 
3 −1
4 1
   
1+i 1 − i
3
(d) x =  1  and y =  −3  in C ,
i −i
   
0 y 1
 0   y 2 
n
.
(e) x =   and y =  .  in .
. .
 .   . 
0 y n

3
5.4.2. Find two vectors of unit norm that are orthogonal to u = .
−2
5.4.3. Consider the following set of three vectors.
1 1 −1
      
 
 
 −1   1   −1 
0 1 2
x 1 =   , x 2 =   , x 3 =   .
 
 
2 0 0
4
(a) Using the standard inner product in , verify that these vec-
tors are mutually orthogonal.
(b) Find a nonzero vector x 4 such that {x 1 , x 2 , x 3 , x 4 } is a set
of mutually orthogonal vectors.
4
(c) Convert the resulting set into an orthonormal basis for .
5.4.4. Using the standard inner product, determine the Fourier expansion of
x with respect to B, where
        
1  1 1 −1 
1 1 1
,
,
x =  0  and B = √  −1  √   √  −1  .
1
 2 3 6 
−2 0 1 2