Question

Prove the set v is the set of positive real numbers with addition and scalar multiplication defined as follows, x y = xy and cx = c c is a vector space.'

Answer 1

The set V is the set of positive real numbers with addition and scalar multiplication defined as x + y = xy and cx = cx, and it can be proven that V is a vector space by satisfying all the vector space axioms.

Step-by-step explanation:

The set V is the set of positive real numbers, denoted as V = (0, +∞).

To prove that V is a vector space, we need to show that it satisfies the vector space axioms:

1. Addition of Vectors: For all x, y ∈ V, x + y = xy is also a positive real number, which belongs to V.
2. Multiplication by Scalar: For all c ∈ ℝ (real numbers) and x ∈ V, cx = cx is also a positive real number, which belongs to V.
3. Associativity of Addition: (x + y) + z = (xy) + z = (xyz) = x + (yz) = x + (y + z)
4. Commutativity of Addition: x + y = y + x
5. Additive Identity: There exists an additive identity element, denoted as 0, such that x + 0 = x for all x ∈ V. In this case, 0 is the number 1 since 1 is the multiplicative identity of the real numbers.
6. Additive Inverse: For every x ∈ V, there exists a unique element -x ∈ V such that x + (-x) = 0.
7. Distributivity of Scalar Multiplication with respect to Vector Addition: c(x + y) = cxy = cxy = cx + cy for all c ∈ ℝ and x, y ∈ V.
8. Distributivity of Scalar Multiplication with respect to Field Addition: (c + d)x = ((cd)x) = cx + dx for all c, d ∈ ℝ and x ∈ V.

Since V satisfies all the vector space axioms, we can conclude that V is a vector space.