1. The problem statement, all variables and given/known data How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? 2. Relevant equations To really prove this rigorously it would make sense to use only the following axioms: 1)For every x,y in V, x+y is also in V. 2) (x+y)+z = x+(y+z) = x+y+z 3) 0 is in V, and 0+x = x for all x in V 4) All x in V have an inverse -x such that x+(-x)=0 5)For all scalars 'a' and 'b', a(bx) = (ab)x 6) For all x in V, 1x=x 7)For all scalars 'a' and 'b', (a+b)x = ax+bx 8)for all scalars 'a', a(x+y) = ax+ay 9)For all x in V, -x = (-1)x 10) A set of N linear independent vectors implies that if a linear combination of them is zero, then all the coefficients are zero. 3. The attempt at a solution I feel like I need to use a proof by contradiction, but not really sure how to start. This is actually my simplification of the true proof, and that is to prove that all vectors can be written as a linear combination of a basis set.