5. Let V ={[x1 x2] such that x1 > 0, x2 < 0; both ∈ R}, with vector addition and scalar multiplication
defined as follows:
[x1 x2] + [y1 y2] = [x1y1 -x2y2]
a[x1 x2] = [x1^a -(-x2)^a]
Check whether V (with the given addition and multiplication) satisfies the axioms
a) a([a b]+[c d])= a[a b]+ a[c d] (axiom S2)
b) (a + b)[a b] = a[a b]+ b[a b] (axiom S3)
defined as follows:
[x1 x2] + [y1 y2] = [x1y1 -x2y2]
a[x1 x2] = [x1^a -(-x2)^a]
Check whether V (with the given addition and multiplication) satisfies the axioms
a) a([a b]+[c d])= a[a b]+ a[c d] (axiom S2)
b) (a + b)[a b] = a[a b]+ b[a b] (axiom S3)
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I'm not sure as to what the additive identity would be, so I can't prove that V is a vector space, but I think I have the answer for showing that the axioms are satisfied:
a([a b] + [c d]) = a([ac -bd]) = [(ac)^a -(bd)^a]
a[a b] + a[c d] = [a^a -(-b)^a] + [c^a -(-d)^a] = [(a^a)(c^a) -{-(-b)^a}{-(-d)^a}] = [(ac)^a -(bd)^a]
(a + b)[a b] = [a^(a + b) -(-b)^(a + b)]
a[a b] + b[a b] = [a^a -(-b)^a] + [a^b -(-b)^b] = [a^(a + b) -(-b)^(a + b)]
a([a b] + [c d]) = a([ac -bd]) = [(ac)^a -(bd)^a]
a[a b] + a[c d] = [a^a -(-b)^a] + [c^a -(-d)^a] = [(a^a)(c^a) -{-(-b)^a}{-(-d)^a}] = [(ac)^a -(bd)^a]
(a + b)[a b] = [a^(a + b) -(-b)^(a + b)]
a[a b] + b[a b] = [a^a -(-b)^a] + [a^b -(-b)^b] = [a^(a + b) -(-b)^(a + b)]