(%i1) load ("linearalgebra");
Warning - you are redefining the Maxima function require_list
Warning - you are redefining the Maxima function matrix_size
Warning - you are redefining the Maxima function rank
(%o1) /usr/local/share/maxima/5.9.2/share/linearalgebra/linearalgebra.mac
(%i2) M : matrix ([1, 2], [1, 2]);
                            [ 1  2 ]
(%o2)                       [      ]
                            [ 1  2 ]
(%i3) nullspace (M);
                               [  1  ]
                               [     ]
(%o3)                     span([   1 ])
                               [ - - ]
                               [   2 ]
(%i4) columnspace (M);
                                [ 1 ]
(%o4)                      span([   ])
                                [ 1 ]
(%i5) ptriangularize (M - z*ident(2), z);
                         [ 1   2 - z   ]
(%o5)                    [             ]
                         [           2 ]
                         [ 0  3 z - z  ]
(%i6) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
                     [ 1 - z    2      3   ]
                     [                     ]
(%o6)                [   4    5 - z    6   ]
                     [                     ]
                     [   7      8    9 - z ]
(%i7) MM : ptriangularize (M, z);
              [ 4  5 - z            6            ]
              [                                  ]
              [                2                 ]
              [     66        z    102 z   132   ]
              [ 0   --      - -- + ----- + ---   ]
(%o7)         [     49        7     49     49    ]
              [                                  ]
              [               3        2         ]
              [           49 z    245 z    147 z ]
              [ 0    0    ----- - ------ - ----- ]
              [            264      88      44   ]
(%i8) algebraic : true;
(%o8)                         true
(%i9) tellrat (MM [3, 3]);
                         3       2
(%o9)                  [z  - 15 z  - 18 z]
(%i10) MM : ratsimp (MM);
               [ 4  5 - z           6           ]
               [                                ]
               [                2               ]
(%o10)         [     66      7 z  - 102 z - 132 ]
               [ 0   --    - ------------------ ]
               [     49              49         ]
               [                                ]
               [ 0    0             0           ]
(%i11) nullspace (MM);
                        [        1         ]
                        [                  ]
                        [   2              ]
                        [  z  - 14 z - 16  ]
                        [  --------------  ]
(%o11)             span([        8         ])
                        [                  ]
                        [    2             ]
                        [   z  - 18 z - 12 ]
                        [ - -------------- ]
                        [         12       ]
(%i12) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]);
                       [ 1   2   3   4  ]
                       [                ]
                       [ 5   6   7   8  ]
(%o12)                 [                ]
                       [ 9   10  11  12 ]
                       [                ]
                       [ 13  14  15  16 ]
(%i13) columnspace (M);
                           [ 1  ]  [ 2  ]
                           [    ]  [    ]
                           [ 5  ]  [ 6  ]
(%o13)                span([    ], [    ])
                           [ 9  ]  [ 10 ]
                           [    ]  [    ]
                           [ 13 ]  [ 14 ]
(%i14) apply ('orthogonal_complement, args (nullspace (transpose (M))));
                           [ 0 ]  [  1  ]
                           [   ]  [     ]
                           [ 1 ]  [  0  ]
(%o14)                span([   ], [     ])
                           [ 2 ]  [ - 1 ]
                           [   ]  [     ]
                           [ 3 ]  [ - 2 ]