- #1
Dustinsfl
- 2,281
- 5
I am not sure if my #4 holds and I don't know how to approach #7. My Axioms are below the general axioms.
{∀ 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
{∀ 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