I am not sure if my #4 holds and I don't know how to approach #7. My Axioms are below the general axioms.(adsbygoogle = window.adsbygoogle || []).push({});

{∀ x ϵ ℝ+ : x>0}

Define the operation of scalar multiplication, denoted ∘, by α∘x = x^α, x ϵ ℝ+ and α ϵ ℝ.

Define the operation of addition, denoted ⊕, by x ⊕ y = x·y, x, y ϵ ℝ+.

Thus, for this system, the scalar product of -3 times 1/2 is given by:

-3∘1/2 = (1/2)^-3 = 8 and the sum of 2 and 5 is given by:

2 ⊕ 5 = 2·5 = 10.

Is ℝ+ a vector space with these operations? Prove your answer.

Vector Space Axioms

1. x + y = y + x

2. (x + y) + z = x + (y + z)

3. x + 0 = x

4. x + (-x) = 0

5. α(x + y) = α·x + α·y

6. (α + β)x = α·x + β·x

7. (αβ)·x = α·(βx)

8. 1·x = x

Axioms:

1. x ⊕ y = x·y = y·x = y ⊕ x

2. (x ⊕ y) ⊕ z = (x·y) ⊕ z = x·y·z = x·(y·z) = x·(y ⊕ z) = x ⊕ (y ⊕ z)

3. x ⊕ 1 = x·1 = x

4. -x = -1∘x = x^-1 = 1/x ⇒x ⊕ (-x) = x·1/x = 1

5. α∘(x ⊕ y) = α(x·y) = (x·y)^α = x^α·y^α = x^α ⊕ y^α = α∘x ⊕ α∘y

6. (α + β)∘x = x^(α + β) = x^α·x^β = x^α ⊕ x^β = α∘x ⊕ β∘x

7. (α·β)∘x =

8. 1∘x = x^1 = x

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# Homework Help: Vector Space {∀ x ϵ ℝ+ : x>0}

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