Problem 4. Let V be a vector space over R. Prove that for any a, b E R and c E V with x ba mplies ах а Hint. Axiom (VS 8) will be needed in your proof.

Definition 0.1. A vector space V over a field F is a set V with and addition operation + and scalar multiplication operation – by elements of F that sasify the following axioms: 1. For all x, y E V, x + y = y +x 2. For all x, y, z E V, (x + y) + z = x + (y z) 3. There exists 0 E V such that x + 0 = x for all x E V. 4. For all E V, there exists -x E V such that x + (-x) = 0 5. For all x E V, 1 x = x. 6. For all a, b e F and x E V, (a + b) -x = a x + b-x. 7. For all a, bE F and x E V, (ab) – 8. For all a e F and x,y e V, a – (x + y) x a (b x). = a x+a – y. The elements of V are called vectors, and it is understood that we write cx to mean c -x. From the axioms above, one can deduce the usual algebraic rules are true in vector spaces: