Let u, v, and w be arbitrary vectors in v. assume u +w=v+w. adding -w to both sides of the equation, we have (u + w) + (-w) = (v + w) + (-w). applying axiom (a2) (the associative law) to both sides of the equation, we have u+(w + (-w)) = v +(w + (- applying axiom (a4) (the law of additive inverses) to both sides of the equation, we have u = v. incorrect the answer above is not correct.
(1 point) in this problem, you will prove the cancellation law of proposition 9.9: theorem. let v be a vector space over some field k. for vectors u, v, w: ifu+w=v+w, then u = v. put 10 of the following sentence fragments in order to form a logically correct proof of the theorem. there is only one correct answer, so be sure not to skip any steps. choose from these: proof of the theorem: u +0=v +0. let u, v, and w be arbitrary vectors in v. u+w=y+w. homework 2 assume u +w=v+w. (-1)(u + w) = (-1)(v + w). adding -w to both sides of the equation, we have (u + w) + (-w) = (v + w) + (-w). multiplying both sides of the equation by -1, we have applying axiom (a2) (the associative law) to both sides of the equation, we have adding w to both sides of the equation, we have assume u = v. u+(w +(-w)) = v +(w + (- applying axiom (a1) (the commutative law) to both sides of the equation, we have applying axiom (a4) (the law of additive inverses) to both sides of the equation, we have
0+u = 0 +v. u = v. applying axiom (a3) (the additive unit law) to both sides of the equation, we have w +u = w +v. 1+w homework 2 +v+w.