Page 344 - 35Linear Algebra
P. 344

344                                                                          Sample Final Exam


                                                                           3
                                  For any special values of a at which U 6= R , express the subspace U as the
                                  span of the least number of vectors possible. Give the dimension of U for
                                                                                  3
                                  these cases and draw a picture showing U inside R .
                               7. Vandermonde determinant: Calculate the following determinants

                                                                                        2    3 
                                                                    2          1  x   x   x
                                                             1 x x
                                                                                       2
                                              1 x                               1  y   y    y  3
                                         det         ,  det   1 y y  2   ,  det       2       .
                                              1 y                    2          1  z   z    z  3
                                                             1 z    z
                                                                                 1 w w   2  w 3
                                  Be sure to factorize you answers, if possible.
                                  Challenging: Compute the determinant

                                                                    2          n−1  
                                                         1 x 1   (x 1 )  · · · (x 1 )
                                                         1 x 2
                                                                (x 2 ) 2  · · · (x 2 )  n−1
                                                                    2              
                                                         1 x 3                       .
                                                                (x 3 )  · · · (x 3 )  n−1
                                                    det 
                                                        .    .      . .
                                                        . .  . .    . .  . .     . 
                                                                                  .
                                                                                  . 
                                                         1 x n (x n ) 2  · · · (x n ) n−1
                                                              
                                                      1       3     1     0      0 
                                                                                                       3
                                                               2
                                                            ,
                                                                        ,
                               8.  (a) Do the vectors                        form a basis for R ?
                                                                  ,
                                                                     0
                                                                                  0
                                                                               ,
                                                         2
                                                                           1
                                                                                    
                                                         3     1     0     0      1
                                       Be sure to justify your answer.
                                                                                           
                                                                                    1         4
                                                                                              3
                                                                                    2        
                                                                                   
                                                        4
                                   (b) Find a basis for R that includes the vectors     and    .
                                                                                    3
                                                                                              2
                                                                                           
                                                                                    4         1
                                   (c) Explain in words how to generalize your computation in part (b) to
                                                         n
                                       obtain a basis for R that includes a given pair of (linearly independent)
                                       vectors u and v.
                               9. Elite NASA engineers determine that if a satellite is placed in orbit starting
                                  at a point O, it will return exactly to that same point after one orbit of the
                                  earth. Unfortunately, if there is a small mistake in the original location of
                                                                                                   1
                                                                                       3
                                  the satellite, which the engineers label by a vector X in R with origin at O,
                              1
                               This is a spy satellite. The exact location of O, the orientation of the coordinate axes
                                3
                            in R and the unit system employed by the engineers are CIA secrets.
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