Let V be the set of all ordered pairs of real numbers and co

Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication op- erations on u = (u1, u2) and v = (v1, v2): u+v=(u1 +v1 +1,u2 +v2 +1), ku=(ku1,ku2)

(a) Compute u+v and ku for u=(0,4),v=(1,3), and k = 2.

(b) Show that (0, 0) = 0.

(c) Show that (1, 1) = 0.

(d) Show that Axiom 5 holds by producing an ordered pair u such that u + (u) = 0 for u = (u1 , u2 ).

(e) Find two vector space axioms that fail to hold.

Solution

a)Given u=(u₁,u₂)=(0,4)

v=(v₁,v₂)=(1,-3)

k=2

so u₁=0,u₂=4,v₁=1,v₂=-3

u+v=(u₁+v₁,u₂+v₂)=(0+1,4-3)=(1,1)

ku=(ku₁,ku₂)=(2*0.2*4)=(0,8)