- #1
Eleni
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Homework Statement
Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar
multiplication of elements of V in an unusual way so that when
u=(x1, y1), v=(x2, y2) and k∈ℝ
u+v= (x1⋅x2, y1+y2) and
k⋅u=(x1/k, y1/k)
Show detailed calculations of one case where V
i) satisfies one of the addition axioms (1 – 5)
ii) fails to uphold one of the addition axioms (1 – 5)
iii) satisfies one of the scalar multiplication axioms (6 – 10)
iv) fails to uphold one of the scalar multiplication axioms (6 – 10)
Homework Equations
The Attempt at a Solution
i)u+v=v+u: Axiom 2
u+v= (x1⋅x2, y1+y2)
v+u= (y1⋅y2, x1+x2)
∴u+v≠v+u
ii) u+(-u)= (-u)+u=(0,0): axiom 4
(x1, y1)+(-x1,-y1)= (x1⋅-x1, y1+-y1)
= (-x1, 0)
iii) 1u=u: axiom 10
ku= (x1/k, y1/k) if k=1
1u= (x1/1, y1/1)= (x1,y1)=u
iv)??
I don't know if I am even on the right track with the three that I have calculated. I feel confident in the third but the others, not so much. I also have struggled to demonstrate the fourth.
Any help would be greatly appreciated.