SOLUTION: Find a basis of the subspace of R4 defined by the equation −5x1−5x2−4x3−8x4=0.

(1)  From general theory, the dimension of this subspace is 4-1 = 3.

     So, we need to find three basic vectors.



(2)  It is enough to find three linearly independent vectors, satisfying the given equation.



(3)  For simplicity, we can use an EQUIVALENT equation instead of the given one

         5x1 + 5x2 + 4x3 + 8x4 = 0.    (1)



(4)  First such vector is  V1 = (1, -1, 0, 0).

     You can check that this vector satisfies equation (1).



(5)  Second such vector is  V2 = (1, 0, -1.25, 0).    The component  -1.25  is  -5/4.

     You can check that this vector satisfies equation (1).



(6)  Third such vector is  V3 = (1, 0, 0, -0.6125).    The component  -0.6125  is  -5/8.

     You can check that this vector satisfies equation (1).



(7)  Finally, it is obvious that vectors V1, V2 and V3 are linearly independent, 
     so vectors V1, V2 and V3 provide the solution to the problem.    ANSWER