Question

Problem #9: Let V be the set of all ordered pairs of real numbers (uj,u) with up > 0. Consider the following addition and scalar multiplication operations on u = (U1, u) and v = (v1, v2): u + v = (U1 + V1 +4,5u2v2), ku = (kuj, kuz) Use the above operations for the following parts. (a) Compute u + v for u =(4,4) and v = (3,2). (b) If the set V satisfies Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector? (c) If u = (6,3), what would be the negative of the vector u referred to in Axiom 5 of a vector space? (Don't forget to use your answer to part (b) here!) Problem #9(a): 11,40 enter your answer in the form a,b Problem #9(b): -4,1/5 enter your answer in the form a,b Problem #9(c): -10,0 enter your answer in the form a,b Just Save Your work has been saved! (Back to Admin Page) Submit Problem #9 for Grading Problem #9 Attempt #1 Attempt #2 Attempt #3 Your Answer: 9(a) 11,40 9a) 9(a) 9(b) -4, 1/5 9(b) 9(b) 9(0) -8,0 9(c) -10,0 9(c) Your Mark: 9(a) 2/2 9(a) 9(a) 9(b) 2/2 9(b) 9(b) 9(C) 0.5/2 vX 9(c) 1/2vx 9(c) Note: Your mark on each question will be the MAXIMUM of your marks on each try. (So there is no harm in making another attempt at a partially correct answer.)

Answer 1

Given: v is the set of all ordered pairs of real numbers. with yo. Addition and scalar multiplication operations on u = (4, 4) and I=(45) are defined as below: ū + v = (u + y +4,54), kū= (ku,, K4₂) (a) =(4,4) F=(3,2) u v = (4+3+4,5(4) (2)). =( 11, 40) u tv = (11,40 (6) Let (5= (nn) be the zero vector then ū+ 0 = ū (4,+4+4, 5ųh) =(4,4₂) → uith, +4=u, 1 n, +4=0 h,=-4 Sum= 4₂ 5 59=1 = i zero vector 5 = (-4, ) (C) ū=66,3) Let v = (V1, V2) be the negative of ū then u + v = ŏ (6+4+4, 5(3/2) = (-4, {::0=(-4,5) (lotv, 1562) = (-4, 5) 10tv, = - 1542 = 3 V=-4-10 2 = 1 3 x 15 V =-14 = 15 :. Negative of u = (-14, 7/5)