1. Consider a matrix A which does not have any zero columns(i.e. at least one entry is non-zero). Is it possible to
reduce this matrix of real numbers  to a matrix with a zero column(i.e. all entries in the column are zero) using
elementary row operation? Prove your answer.
                                                                                                                            
2. Decide the dependence or independence of
        a) (1,1,2), (1,2,1), (3,1,1);
        b) v_1 - v_2 , v_2 - v_3 , v_3 - v_4 , v_4 - v_1 for any vectors v_1, v_2, v_3, v_4;
        c) (1,1,0), (1,0,0) (0,1,1), (x,y,z) for any number x, y, z.

3. Explain why Ax = b is solvable if and only if rank A = rank A` where A` is formed  from A by adding b as an extra
column (augmented matrix). The rank is the dimension of the vector space formed by columns of matrix (column space).
                                                                                                                            
4.Given points v_1, v_2 and v_3, show that every point in the interior of the triangle defined by these three points
including the boundaries is
given by
                a_1*v_1 + a_2*v_2 + a_3*v_3
such that
                0 <= a_i <= 1
         and a_1 + a_2 + a_3 = 1.

5 Determine if the following system is consistent or inconsistent.
				3x + 4y + z = 1
				2x + 3y		= 0
				4x + 3y - z  = -2

6 Solve graphically & find region of intersection of following system
				2x + 3y <= 6
				6x - 5y  >= 0
				2x - 5y  <= 7