Page 131 - 35Linear Algebra
P. 131

7.2 Review Problems                                                                           131


                          (c) Find the matrix for  d  acting on the vector space V in the ordered
                                                  dx
                                     0
                                                   2
                                           2
                              basis B = (x + x, x − x, 1).
                         (d) Use the matrix from part (c) to rewrite the differential equation
                               d  p(x) = x as a matrix equation. Find all solutions of the matrix
                              dx
                              equation. Translate them into elements of V .


                          (e) Compare and contrast your results from parts (b) and (d).

                      4. Find the “matrix” for  d  acting on the vector space of all power series
                                                dx
                                                        3
                                                     2
                         in the ordered basis (1, x, x , x , ...). Use this matrix to find all power
                         series solutions to the differential equation  d  f(x) = x. Hint: your
                                                                       dx
                         “matrix” may not have finite size.

                      5. Find the matrix for  d 2 2 acting on {c 1 cos(x) + c 2 sin(x) | c 1 , c 2 ∈ R} in
                                              dx
                         the ordered basis (cos(x), sin(x)).


                      6. Find the matrix for  d  acting on {c 1 cosh(x) + c 2 sinh(x)|c 1 , c 2 ∈ R} in
                                              dx
                         the ordered basis
                                                    (cosh(x), sinh(x))
                         and in the ordered basis

                                         (cosh(x) + sinh(x), cosh(x) − sinh(x)).





                                         2
                      7. Let B = (1, x, x ) be an ordered basis for
                                                              2
                                         V = {a 0 + a 1 x + a 2 x | a 0 , a 1 , a 2 ∈ R} ,
                                  0
                                        3
                                            2
                         and let B = (x , x , x, 1) be an ordered basis for
                                                                3
                                                         2
                                   W = {a 0 + a 1 x + a 2 x + a 3 x | a 0 , a 1 , a 2 , a 3 ∈ R} ,
                         Find the matrix for the operator I : V → W defined by
                                                            Z  x
                                                   Ip(x) =      p(t)dt
                                                              1
                         relative to these bases.


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