Let V be the set of all ordered pairs of real numbers ( u 1 ,? u 2 ) with u 2 >

Question

Let V be the set of all ordered pairs of real numbers (u1,?u2) with u2 > 0. Consider the following addition and scalar multiplication operations on u = (u1,?u2) and v = (v1,?v2):

u + v = (u1 + v1 + 4,?7u2v2),     ku = (ku1,?ku2)

Use the above operations for the following parts.

If the set V satisfies Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector?

If u = (6,?8), 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!)

Let V be the set of all ordered pairs of real numbers (u1,?u2) with u2 > 0. Consider the following addition and scalar multiplication operations on u = (u1,?u2) and v = (v1,?v2):

u + v = (u1 + v1 + 4,?7u2v2),     ku = (ku1,?ku2)

Use the above operations for the following parts.

If the set V satisfies Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector?

If u = (6,?8), 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!)

Explanation / Answer

According to the given problem,

(a) The vector 0=(-4; 1/7) is the zero vector since:
u+0 = (u1-4 + 4 ; 7.u2/7) = (u1;u2) = u
And likewise, 0+v = v

(b) Let's find v such that: u + v = 0
u+v = 0
(u1 + v1 + 4,?7u2v2) = (-4; 1/7)
(v1,v2) = (-u1 - 8; 1/(49×u2))

In the case where u=(6,8), the additive inverse of u is:
-u = (-14 ; 1/392)

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