Page 275 - 35Linear Algebra
P. 275

14.7 Review Problems                                                                          275


                      7. (a) Show that if Q is an orthogonal n × n matrix, then

                                                     u v = (Qu) (Qv) ,

                                              n
                              for any u, v ∈ R . That is, Q preserves the inner product.
                         (b) Does Q preserve the outer product?
                          (c) If the set of vectors {u 1 , . . . , u n } is orthonormal and {λ 1 , · · · , λ n }
                              is a set of numbers, then what are the eigenvalues and eigenvectors
                                                  P n        T
                              of the matrix M =         λ i u i u ?
                                                    i=1      i
                         (d) How would the eigenvectors and eigenvalues of this matrix change
                              if we replaced {u 1 , . . . , u n } by {Qu 1 , . . . , Qu n }?

                      8. Carefully write out the Gram-Schmidt procedure for the set of vectors

                                                                     
                                                         
                                               1          1        1 
                                                   1  ,  −1    ,    1     .
                                                                
                                                   1       1      −1
                                                                      
                         Is it possible to rescale the second vector obtained in the procedure to
                         a vector with integer components?

                      9. (a) Suppose u and v are linearly independent. Show that u and v       ⊥
                                                                                ⊥
                              are also linearly independent. Explain why {u, v } is a basis for
                              span{u, v}.


                                                             Hint


                         (b) Repeat the previous problem, but with three independent vectors
                              u, v, w where v ⊥  and w ⊥  are as defined by the Gram-Schmidt
                              procedure.

                    10. Find the QR factorization of

                                                                    
                                                          1    0    2
                                                 M =   −1     2    0   .
                                                        −1 −2       2

                                                                   ⊥
                                                                         ⊥
                    11. Given any three vectors u, v, w, when do v or w of the Gram–Schmidt
                         procedure vanish?

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