Page 274 - 35Linear Algebra
P. 274

274                                                      Orthonormal Bases and Complements


                                       from the solution set of the matrix equation

                                                                     T
                                                                   v x = 0 .
                                                                     1
                                       Pick the vector v 2 obtained from the standard Gaussian elimina-
                                       tion procedure which is the coefficient of x 2 .
                                  (b) Pick a vector perpendicular to both v 1 and v 2 from the solutions
                                       set of the matrix equation
                                                                    T
                                                                   v 1  x = 0 .
                                                                   v T
                                                                    2
                                       Pick the vector v 3 obtained from the standard Gaussian elimina-
                                       tion procedure with x 3 as the coefficient.
                                   (c) Pick a vector perpendicular to v 1 , v 2 , and v 3 from the solution set
                                       of the matrix equation
                                                                     
                                                                   v 1 T
                                                                     T
                                                                 v  x = 0 .
                                                                      
                                                                 
                                                                    2
                                                                   v 3 T
                                       Pick the vector v 4 obtained from the standard Gaussian elimina-
                                       tion procedure with x 3 as the coefficient.
                                  (d) Normalize the four vectors obtained above.

                               5. Use the inner product
                                                                    1
                                                                  Z
                                                          f · g :=    f(x)g(x)dx
                                                                   0
                                                                      2
                                                                         3
                                  on the vector space V = span{1, x, x , x } to perform the Gram-Schmidt
                                                                         2
                                                                            3
                                  procedure on the set of vectors {1, x, x , x }.
                               6. Use the inner product
                                                                 Z  2π
                                                         f · g :=     f(x)g(x)dx
                                                                   0
                                  on the vector space V = span{sin(x), sin(2x), sin(3x)} to perform the
                                  Gram-Schmidt procedure on the set of vectors {sin(x), sin(2x), sin(3x)}.
                                  Try to build an orthonormal basis for the vector space

                                                          span{sin(nx) | n ∈ N} .


                                                      274
   269   270   271   272   273   274   275   276   277   278   279