Is ℝ+ a Vector Space with Scalar Multiplication and Addition?

In summary, Axioms 1-3 imply that the scalar product of two vectors is the sum of the vectors, while Axiom 4 states that the product of two vectors is the negative of the sum of the vectors. Axiom 5 states that the product of two scalars is the scalar product of the first vector with the second scalar. Axiom 6 states that the product of two vectors is the vector product of the first vector with the transpose of the second vector. Axiom 7 states that the product of two vectors is the vector product of the first vector with the complex conjugate of the second vector. Finally, Axiom 8 states that the product of two vectors is the sum of the components of the
  • #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
 
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  • #2
For #4, you can just note that 1/x is in R+, and that it adds with x to give the identity element. You really don't have to show how you figured out what the additive inverse is, just that it's in the space.

For #7, if you can't see it starting from the lefthand side, try seeing what happens if you start with the righthand side.
 
  • #3
7. (α·β)∘x = x^(α·β) = x^(β·α) = (β∘x)^α = α∘(β∘x)
So this is what I obtained. Is it correct?
 
  • #4
I'd probably add one step:

[tex]x^{\beta\alpha} = (x^\beta)^\alpha = (\beta\circ x)^\alpha[/tex]
 
  • #5
Do you know how to do superscript in Maple with the greeks, by any chance? I can only do it with standard letters.
 
  • #6
Here is the completed Vector Space problem: Can you tell me if you see any issues? Thanks.
{∀ 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.

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 = x^(α·β) = x^(β·α) = (x^β)^α = (β∘x)^α = α∘(β∘x)
8. 1∘x = x^1 = x

Closure Properties:
1. If x ϵ ℝ+ and α is a scalar, then α∘x ϵ ℝ+.
α∘x = x^α > 0 ∴ α∘x ϵ ℝ+
2. If x,y ϵ ℝ+, then x ⊕ y ϵ ℝ+.
x ⊕ y = x·y > 0 ∴ x ⊕ y ϵ ℝ+

Yes, ℝ+ is a vector with these operations.
 
  • #7
Dustinsfl said:
Do you know how to do superscript in Maple with the greeks, by any chance? I can only do it with standard letters.
Nope, sorry. I've never used Maple. Someone else can probably answer your question, hopefully.
 
  • #8
Hi, I don't mean to necro topics but I just have a quick question on the same title:

For the above Vector Space {∀ x ϵ ℝ+ : x>0} to be a vector space, is it necessary to contain the zero vector? (Which it doesn't, since x>0 so I'm thinking it's not a vector space).
 
  • #9
The zero vector is the identity element for vector addition. What it is depends on your definition for vector addition. It doesn't have to actually equal 0, the identity element of the usual addition of real numbers.
 
  • #10
It is necessary that a vector space be non-empty. But once you have a vector, v, in the vector space, (-1)v= -v is in the set and so v+ (-v)= 0 is in the set. So just saying that the vector space is non-empty is equivalent to saying that it contains the 0 vector.
 
  • #11
Hmm I might be missing the point here, but say we use ordinary addition and scalar multiplication. My question is that for a VS defined to be only positive reals, there is no number you can add to a number x such that you obtain x. So there is no zero element since zero is non positive (not in the set) so R+ is not a vector space!
 
  • #12
Yes, with regular addition and multiplication, the set of positive reals does not satisfy the requirements of a vector space. But that's not the question that was originally asked in this thread.

The point you seem to be missing is that a vector space isn't simply a set. It's a set and two binary operators which define scalar multiplication and vector addition. x=1 is the "zero" vector, the identity element for vector addition as it was defined in this problem. R+ and the given operators do satisfy the axioms of a vector space.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on these vectors. It is used to model various physical and abstract concepts in science and engineering.

2. What does the notation "∀ x ϵ ℝ+ : x>0" mean?

The notation "∀ x ϵ ℝ+ : x>0" means "for all elements x in the set of positive real numbers, x is greater than 0." In other words, it represents the condition that all elements in the vector space must be positive real numbers.

3. How is a vector space different from a Euclidean space?

A Euclidean space is a specific type of vector space that is defined by the properties of Euclidean geometry, such as distance and angle. A vector space, on the other hand, is a more general concept that can have different sets of operations and properties, depending on its application.

4. How is a vector space used in science?

Vector spaces are used in various fields of science, such as physics, engineering, and computer science, to model and analyze physical quantities and their relationships. They are also used in data analysis and machine learning to represent and manipulate data.

5. What are some examples of vector spaces?

Some examples of vector spaces include the set of all real numbers, the set of all 2-dimensional vectors, and the set of all polynomials of degree less than or equal to n. Other examples can be found in fields such as quantum mechanics, where vector spaces are used to represent states of physical systems.

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