Question

Question 1 (4 Marks) A weird vector space. Consider the set R+ = {2 ER: I >0} = V. We define addition by zey=ry, the product of x and y. We use the field F=R, and define multiplication by cor = xº. Prove that (V, e, Ro) is a vector space. ONLY HAND IN : i) The zero vector ii) what is 6-7 iii) proof of e) of the axioms.

Answer 1

Since I don't know which is the e^{​​​​​​th} axiom in your notes/book I have proved each vector space axiom. Verify accordingly

प let v = (Rt, 0, 0 where 18 19 А o cox: х" А) Abelian group The main question (y, 0 0) "u alects. Арал 94 у 4 closure Lek хуу° - X, 8 70 Хей: 28 27 0 4 97 от 1870 x ®я 7 0 o holds closure since А 2) & Associative since product of positive real nos. is associative (a® W5 хч) 3- 192 20 (3) = х 45 = ху, 3) The zero vector (identity element) XEV be such that xey: 8€х - yvyev let ХФ 9- 1 19-9 х = 1 vector the zero и 1

& Inverse element. Let atv & bev such that a b 1 = be a = a b = 7 ab=1 b-1/a for each aev -a = 7 where ta represent the invesse of a. 5 commutation Let a YEV a xey = xy = yx=yor commutative (V) is an Abulian group B Let XEV & CER Since XEV 9 x70 for each CER x > 0 Cox = x 70 COX EV c) Let XyEV & CER 470, 470 I co cx@y)e co cxy) (xly) sinca x, y 70 x 70 Gay) - co (my) EV 70

Date: / / 2 Let XEV A c, d E R (c +d30 yerd=6xxd) =ODO dox) C, DER & NGV co(dox) = colad) (xas = xed (cdox E Let F) Let CER such that cox= x= xe=x e-1 :: multiplicative identity is 1 (V, 6 o is a vector space second part in the zero vector is : 1 ERt. 6-726 € (-7) 16-Y 6 (1) 60 (97)6x now 7