> with(LinearAlgebra):
> A := <<1,1,1,1>|<1,-2,1,0>|<0,0,0,1>>;

                              [1     1    0]
                              [            ]
                              [1    -2    0]
                         A := [            ]
                              [1     1    0]
                              [            ]
                              [1     0    1]

> colsA := Column(A, 1..3);

                                [1]  [ 1]  [0]
                                [ ]  [  ]  [ ]
                                [1]  [-2]  [0]
                       colsA := [ ], [  ], [ ]
                                [1]  [ 1]  [0]
                                [ ]  [  ]  [ ]
                                [1]  [ 0]  [1]

> lst_colsA:= [ colsA ];

                                  [1]  [ 1]  [0]
                                  [ ]  [  ]  [ ]
                                  [1]  [-2]  [0]
                    lst_colsA := [[ ], [  ], [ ]]
                                  [1]  [ 1]  [0]
                                  [ ]  [  ]  [ ]
                                  [1]  [ 0]  [1]

> GramSchmidt(lst_colsA);

                          [1]  [ 1]  [-1/4]
                          [ ]  [  ]  [    ]
                          [1]  [-2]  [-1/4]
                         [[ ], [  ], [    ]]
                          [1]  [ 1]  [-1/4]
                          [ ]  [  ]  [    ]
                          [1]  [ 0]  [3/4 ]

> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> LsqPkg/InitPkg.mpl");

  libname := /afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlg\

        ebra/Maple, "/afs/bp/dist/maple60/lib"

> with(LsqPkg);

                            [Gram_Schmidt]

> printlevel:=3:
> Gram_Schmidt(A);

                                     [1]
                                     [ ]
                                     [1]
                              v[1]=, [ ]
                                     [1]
                                     [ ]
                                     [1]


                                     [1]
                                     [ ]
                                     [1]
                              u[1]=, [ ]
                                     [1]
                                     [ ]
                                     [1]


                           ||u[1]||^2 = , 4


                                    [ 1]
                                    [  ]
                                    [-2]
                             v[2]=, [  ]
                                    [ 1]
                                    [  ]
                                    [ 0]


                                    [ 1]
                                    [  ]
                                    [-2]
                             u[2]=, [  ]
                                    [ 1]
                                    [  ]
                                    [ 0]


                           <u[1], u[2]>=, 0


                           ||u[2]||^2 = , 6


                                     [0]
                                     [ ]
                                     [0]
                              v[3]=, [ ]
                                     [0]
                                     [ ]
                                     [1]


                                   [-1/4]
                                   [    ]
                                   [-1/4]
                            u[3]=, [    ]
                                   [-1/4]
                                   [    ]
                                   [3/4 ]


                           <u[1], u[3]>=, 0


                           <u[2], u[3]>=, 0


                          ||u[3]||^2 = , 3/4


                          [1     1    -1/4]
                          [               ]
                          [1    -2    -1/4]
                          [               ]
                          [1     1    -1/4]
                          [               ]
                          [1     0    3/4 ]

> printlevel:=0:
> Q := Gram_Schmidt(A, 'R');

                             [1     1    -1/4]
                             [               ]
                             [1    -2    -1/4]
                        Q := [               ]
                             [1     1    -1/4]
                             [               ]
                             [1     0    3/4 ]

> R;

                           [1    0    1/4]
                           [             ]
                           [0    1     0 ]
                           [             ]
                           [0    0     1 ]

> Q . R;

                            [1     1    0]
                            [            ]
                            [1    -2    0]
                            [            ]
                            [1     1    0]
                            [            ]
                            [1     0    1]

> ?Matrix
> ?RandomMatrix
> B := RandomMatrix(4,4);

                         [ 23     38    55    -65]
                         [                       ]
                         [ 25     97    68      5]
                    B := [                       ]
                         [  5    -82    26     66]
                         [                       ]
                         [-75    -66    13    -36]

> Gram_Schmidt(B);

            [        8695     438259765    -815187374861]
            [ 23     ----     ---------    -------------]
            [        756       9753579      13680324914 ]
            [                                           ]
            [       51557     38138740     274878902338 ]
            [ 25    -----     --------     ------------ ]
            [        756       886689       6840162457  ]
            [                                           ]
            [       -66347    454258726    774135057094 ]
            [  5    ------    ---------    ------------ ]
            [        756       9753579     34200812285  ]
            [                                           ]
            [        5143     101508541    -230472959381]
            [-75     ----     ---------    -------------]
            [        252       3251193      68401624570 ]

>