Given the definition of a linear manifold as a set M of undefined elements "X" and scalars "a" from a real field F; and an association X+Y between elemets of M, and an association aX between elements F and M, such that:

X+Y = Y+X

a(X+Y) = aX + aY

etc

How do you prove the existence of elements of M for which aX + bY + cZ +.. = 0 has no solution other than a = b = c = ... = 0. Or, more basically, elements X and Y such that aX +bY= 0 has no solution other than a = 0 and b= 0.