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

Recall a previous posting [and classwork]:

/--------------------------------------------------------\
  /----- Method 2 ------------------------------------------\
  An alternative method to compute a basis for ColSpan(S):

    Put the columns of S as rows of a matrix T; i.e, let T be
  S-transpose.  Let b_1,...,b_{97} denote the rows of T.

  Define B := RREF(T).   Note that even though T is
  S-transpose, this B is 
  almost-certainly /not/ the transpose of RREF(S).

  Suppose rows

	  q_7, q_{23}, q_{35}.

  are the pivot-rows of B.  Then (this takes a moment's
  thought) collection

	  {q_7, q_{23}, q_{35}}

  is a basis for RowSpan(B).  But

	  RowSpan(B) = RowSpan(T) = ColSpan(S).
  \________________________________________________________/
\________________________________________________________/




                on Thu Oct 14 18:58:38 EDT 2010


> 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]
So

		  [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]


> I3 := eye(3); onerow := <0|0|0> : threecol := <I3,onerow>;
				[1    0    0]
				[           ]
			  I3 := [0    1    0]
				[           ]
				[0    0    1]

				[1    0    0]
				[           ]
				[0    1    0]
		    threecol := [           ]
				[0    0    1]
				[           ]
				[0    0    0]


> SWithRsPivots := <S | threecol> ;

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

>  RREF(SWithRsPivots) ;
	   [0    1    2    0    0    2    1    0    -1    0]
	   [0    0    0    1    0    1    0    0     3    1]
	   [0    0    0    0    1    0    2    0     1    1]
	   [0    0    0    0    0    0    0    1     1    0]
					       ^     ^
					       ^     ^
> T := Tpose(S);
                               [0     0     0     0]
                               [1     1    -1     2]
                               [2     2    -2     4]
                          T := [0     1    -1     1]
                               [0    -1     2    -1]
                               [2     3    -3     5]
                               [1    -1     3     0]

> RREFTposeS := RREF(T);
                                     [1    0    0    1]
                                     [0    1    0    1]
                                     [0    0    1    0]
                       RREFTposeS := [0    0    0    0]
                                     [0    0    0    0]
                                     [0    0    0    0]
                                     [0    0    0    0]

> b1 := Tpose(RREFTposeS[1,1..4]);
                                         [1]
                                   b1 := [0]
                                         [0]
                                         [1]

> b2 := Tpose(RREFTposeS[2,1..4]); b3 := Tpose(RREFTposeS[3,1..4]);

                                         [0]
                                   b2 := [1]
                                         [0]
                                         [1]

                                         [0]
                                   b3 := [0]
                                         [1]
                                         [0]

> BThenS := <b1 | b2 | b3 | S >;

	      [1    0    0    0     1     2     0     0     2     1]
     BThenS = [0    1    0    0     1     2     1    -1     3    -1]
	      [0    0    1    0    -1    -2    -1     2    -3     3]
	      [1    1    0    0     2     4     1    -1     5     0]

> RREF(BThenS);
	      [1    0    0    0     1     2     0     0     2     1]
	      [0    1    0    0     1     2     1    -1     3    -1]
	      [0    0    1    0    -1    -2    -1     2    -3     3]
	      [0    0    0    0     0     0     0     0     0     0]