Page 276 - 35Linear Algebra
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276                                                      Orthonormal Bases and Complements


                             12. For U a subspace of W, use the subspace theorem to check that U      ⊥  is
                                  a subspace of W.

                             13. Let S n and A n define the space of n×n symmetric and anti-symmetric
                                                                                                      n
                                  matrices, respectively. These are subspaces of the vector space M of
                                                                                                     n
                                                                      n
                                  all n × n matrices. What is dim M , dim S n , and dim A n ? Show that
                                                                      n
                                    n
                                  M = S n + A n . Define an inner product on square matrices
                                    n
                                                             M · N = trMN .
                                                    n
                                      ⊥
                                  Is A = S n ? Is M = S n ⊕ A n ?
                                      n             n
                             14. The vector space V = span{sin(t), sin(2t), sin(3t), sin(3t)} has an inner
                                  product:
                                                                  Z  2π
                                                          f · g :=    f(t)g(t)dt .
                                                                   0
                                  Find the orthogonal compliment to U = span{sin(t) + sin(2t)} in V .
                                                                                               ⊥
                                  Express sin(t) − sin(2t) as the sum of vectors from U and U .






































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