gistfile1.txt

(%i1) t: matrix([%t^3],[%t^2],[%t],[1])
;
                                                                              [   3 ]
                                                                              [ %t  ]
                                                                              [     ]
                                                                              [   2 ]
(%o1)                                                                         [ %t  ]
                                                                              [     ]
                                                                              [ %t  ]
                                                                              [     ]
                                                                              [  1  ]
(%i2) m: matrix([-1, 3, -3, 1], [3, -6, 3, 0], [-3, 3, 0, 0], [1, 0, 0, 0]);
                                                                       [ - 1   3   - 3  1 ]
                                                                       [                  ]
                                                                       [  3   - 6   3   0 ]
(%o2)                                                                  [                  ]
                                                                       [ - 3   3    0   0 ]
                                                                       [                  ]
                                                                       [  1    0    0   0 ]
(%i3) c: matrix([a, c2, c3, b]);
(%o3)                                                                    [ a  c2  c3  b ]
(%i4) transpose(t) . m . c;
                                              2                         3
(%o4)                                       %t  (3 c3 - 6 c2 + 3 a) + %t  (- 3 c3 + 3 c2 + b - a) + %t (3 c2 - 3 a) + a
(%i5) solve(transpose(t) . m . c, c2);
                                                             3       2         3        3       2
                                                        (3 %t  - 3 %t ) c3 - %t  b + (%t  - 3 %t  + 3 %t - 1) a
(%o5)                                             [c2 = -------------------------------------------------------]
                                                                             3       2
                                                                         3 %t  - 6 %t  + 3 %t
(%i6) solve(transpose(t) . m . c, c3);
                                                         3       2                3          3       2
                                                    (3 %t  - 6 %t  + 3 %t) c2 + %t  b + (- %t  + 3 %t  - 3 %t + 1) a
(%o6)                                         [c3 = ----------------------------------------------------------------]
                                                                                 3       2
                                                                             3 %t  - 3 %t
(%i7)