Problem (C2).  Here we are finding a basis for Range(L_S), 
i.e, a basis for ColSpan(S).

Recall a previous posting [and classwork]:

/--------------------------------------------------------\
  Let c_1, ..., c_{97} denote the columns of S.  Let

	  R := RREF(S).

  Suppose the pivot-cols of R have indices 2, 5 and 19.

  /----- Method 1 ------------------------------------------\
  To compute a basis for ColSpan(S), use the same col-numbers as
  give a basis for R.  Thus collection

	  {c_2, c_5, c_{19}}

  is a basis for ColSpan(S).  After all, row-equiv doesn't
  change lin-relations among cols.
  \________________________________________________________/
\________________________________________________________/


 date 
  @poincare.math.ufl.edu:	START of interactive part of "~/Top/jk.cshrc" 
 unsetenv DISPLAY 
		on Tue Oct 19 12:10:32 EDT 2010

                          "jk: END of    0.mapleinit"



> S := <<0|1|2|0|0|2|1>,<0|1|2|1|-1|3|-1>,<0|-1|-2|-1|2|-3|3>,<0|2|4|1|-1|5|0>>; R := RREF(S);

                      [0     1     2     0     0     2     1]
                      [                                     ]
                      [0     1     2     1    -1     3    -1]
                 S := [                                     ]
                      [0    -1    -2    -1     2    -3     3]
                      [                                     ]
                      [0     2     4     1    -1     5     0]

                      [0     1     2     0     0     2     1]
                      [                                     ]
                      [0     0     0     1     0     1     0]
                 R := [                                     ]
                      [0     0     0     0     1     0     2]
                      [                                     ]
                      [0     0     0     0     0     0     0]

%% Matrix R (which is in RREF) has pivot column-indices 2,4,5.
%% Hence the corresponding columns in S form a
%% basis for the column-span of S.

> s2 := S[1..4,2]; s4 := S[1..4,4]; s5 := S[1..4,5];

                                        [ 1]
                                        [  ]
                                        [ 1]
                                  s2 := [  ]
                                        [-1]
                                        [  ]
                                        [ 2]

                                        [ 0]
                                        [  ]
                                        [ 1]
                                  s4 := [  ]
                                        [-1]
                                        [  ]
                                        [ 1]

                                        [ 0]
                                        [  ]
                                        [-1]
                                  s5 := [  ]
                                        [ 2]
                                        [  ]
                                        [-1]


> ScolsThenS := <s2 | s4 | s5 | S>;
                  [ 1     0     0    0     1     2     0     0     2     1]
                  [                                                       ]
                  [ 1     1    -1    0     1     2     1    -1     3    -1]
    ScolsThenS := [                                                       ]
                  [-1    -1     2    0    -1    -2    -1     2    -3     3]
                  [                                                       ]
                  [ 2     1    -1    0     2     4     1    -1     5     0]

%% We expect that there will be /no/ pivot-cols in the S-copy, since
%% we've made the first 3 cols to be a basis for the colspan of S.


> RREF(ScolsThenS);
               [1    0    0    0    1    2    0    0    2    1]
               [                                              ]
               [0    1    0    0    0    0    1    0    1    0]
               [                                              ]
               [0    0    1    0    0    0    0    1    0    2]
               [                                              ]
               [0    0    0    0    0    0    0    0    0    0]

%% As expected...



> SThenScols := <S | s2 | s4 | s5>;
                  [0     1     2     0     0     2     1     1     0     0]
                  [                                                       ]
                  [0     1     2     1    -1     3    -1     1     1    -1]
    SThenScols := [                                                       ]
                  [0    -1    -2    -1     2    -3     3    -1    -1     2]
                  [                                                       ]
                  [0     2     4     1    -1     5     0     2     1    -1]


> RREF(SThenScols);
               [0    1    2    0    0    2    1    1    0    0]
               [                                              ]
               [0    0    0    1    0    1    0    0    1    0]
               [                                              ]
               [0    0    0    0    1    0    2    0    0    1]
               [                                              ]
               [0    0    0    0    0    0    0    0    0    0]



::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
>  s7 := S[1..4,7]; 
                                        [ 1]
                                        [  ]
                                        [-1]
                                  s7 := [  ]
                                        [ 3]
                                        [  ]
                                        [ 0]

> ScolsThenS := <s7 | s2 | s4 |  S>;  RREF(ScolsThenS);
                  [ 1     1     0    0     1     2     0     0     2     1]
                  [                                                       ]
                  [-1     1     1    0     1     2     1    -1     3    -1]
    ScolsThenS := [                                                       ]
                  [ 3    -1    -1    0    -1    -2    -1     2    -3     3]
                  [                                                       ]
                  [ 0     2     1    0     2     4     1    -1     5     0]

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

> SThenScols := <S | s4 | s7 | s2 >;  RREF(SThenScols) ;
                  [0     1     2     0     0     2     1     0     1     1]
                  [                                                       ]
                  [0     1     2     1    -1     3    -1     1    -1     1]
    SThenScols := [                                                       ]
                  [0    -1    -2    -1     2    -3     3    -1     3    -1]
                  [                                                       ]
                  [0     2     4     1    -1     5     0     1     0     2]

                  [0    1    2    0    0    2    1    0    1    1]
                  [                                              ]
                  [0    0    0    1    0    1    0    1    0    0]
RREF(SThenScols) =[                                              ]
                  [0    0    0    0    1    0    2    0    2    0]
                  [                                              ]
                  [0    0    0    0    0    0    0    0    0    0]