text: find all x = [x,y,z]  where ax = 0? a = [ 1, -1, -2       2, -2, -4       3, -3, -6 ] do the matrix vector multiplication and demand ax=0. then 1x - 1y - 2z = 0 (eq.1) 2x - 2y - 4z = 0 (eq.2) 3x - 3y - 4z = 0 the first and last equations are identical. so you have only two equations. from (1) x = y + 2z  gives in (2) 2(y + 2z) -2y - 4z = 0 this is true for all y and z. now the solution is x = [y + 2z . y, z] in which y and z can be chosen as you like it.