Page 364 - 35Linear Algebra
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364                                                                          Sample Final Exam


                                   (g) Just use the matrix recursion relation of part (d) repeatedly:

                                                                                       n
                                                                       2
                                                    X n+1 = MX n = M X n−1 = · · · = M X 1 .
                                                                 √                 √
                                   (h) The eigenvalues are ϕ =  1+ 5  and 1 − ϕ =  1− 5 .
                                                                 2                2
                                   (i)

                                                              F n+1       n          n  −1
                                                    X n+1 =          = M X n = PD P       X 1
                                                               F n
                                                         n     1    !                               1  !
                                                            √                n               √
                                               ϕ     0             ?   1         ϕ       0
                                         = P                   1 5         = P              n       1 5
                                               0 1 − ϕ      − √    ?   0         0   (1 − ϕ)     − √
                                                                5                                   5
                                                        √
                                                              √ !       ϕ n  !              !
                                                     1+ 5   1− 5        √              ?
                                                                         5
                                                 =     2      2        (1−ϕ) n  =  ϕ −(1−ϕ) n  .
                                                                                    n
                                                       1      1      −  √              √
                                                                          5             5
                                       Hence
                                                                     n
                                                                    ϕ − (1 − ϕ) n
                                                              F n =      √        .
                                                                           5
                                       These are the famous Fibonacci numbers.
                              15. Call the three vectors u, v and w, respectively. Then
                                                                                   1
                                                                                   
                                                                                   4
                                                             u v          3     −  3
                                                     ⊥
                                                    v = v −                       4 
                                                                                     ,
                                                             u u  u = v − u =     1
                                                                          4
                                                                                   4  
                                                                               
                                                                                   1
                                                                                   4
                                  and
                                                                                              
                                                                                             −1
                                                    u w       v ⊥  w          3     3        0 
                                                                     ⊥
                                                                                      ⊥
                                           ⊥
                                          w = w −        u −        v = w − u −     4  v =    
                                                    u u      v ⊥  v ⊥         4     3        0 
                                                                                    4
                                                                                              1
                                  Dividing by lengths, an orthonormal basis for span{u, v, w} is
                                                           1        3       √ 
                                                               √            
                                                           2
                                                                          −  2 
                                                                   6            
                                                                            2   
                                                               √            
                                                                             
                                                        1    −  3    0    
                                                                   √  ,          .
                                                          2      2 
                                                           , 
                                                                           0  
                                                        1       3           
                                                                             
                                                                   √
                                                        2       6       √   
                                                                                
                                                           1        3
                                                                                
                                                                             2 
                                                           2        6        2
                              16.  (a) The columns of M span imL in the basis given for W.
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