Math 2602 B1,B2                                                                            Homework # 10
                                         Due Thursday, March 31, 2005


        Let Ax = b be the matrix/vector form of the following linear system:

               2x + 3y + 6z + 3w  =  2
               2x + 3y + 4z + 4w  =  2
               6x + 9y + 14z + 11w = 6

        i)  Find all solutions to this system.

        ii)  Find a basis for the null space of  A  (the subspace of all solutions to  Ax = 0 ).

        iii)  The vector ( 1 , 0 , 1 , -1 )^T is a solution to Ax = b , where b = ( 5 , 2 , 9 )^T .  Find all
               solutions in this case.










  The remaining parts are for discussion only, and will not be graded.

        iv)  Find all solutions with x = y = 1.

         v)  Are there any solutions with x = y = z ?

   The column space of A is the vector subspace of all linear combinations of columns of  A . The
    original columns of  A  which correspond to "pivot" variables (the "pivot columns") are a basis for
    the column space.

         vi)  What is the dimension of the column space of  A ?

         vii)  Show that each column of  A  is a linear combination of the pivot columns.

    There is a solution of  Ax = b if and only if  b is in the column space of A (is a linear combination
    of the columns of  A) .

 
        viii)  Why can we tell from the answer to (vi) that  there are vectors b for which Ax = b does not
                 have a solution?