Question: Let B1;B2 be given column vectors in Rm and suppose that the set {B1,B2} is independent. Suppose that A is an mxn matrix and that AX1 = B1, AX2 = B2 for some column vectors X1;X2 in Rn. Show that {X1,X2} is independent.

Thought:well you need to show that c1x1 + c2x2 = 0 --> c1 = c2 = 0. Suppose that c1x1 + c2x2 = 0. Then A(c1x2 + c2x2) = c1B1 + c2B2 = 0,because A(0) = 0 for any matrix A.

since{B1, B2} is independent, c1 = c2 = 0. Thus {x1,x2} is independent as well.

Is this correct?