Question

Let \(\mathbb{R}^{+}\) denote the set of positive real numbers. Define the "sum" of two elements of \(\mathbb{R}^{+}\) to be their usual product, and define scalar multiplication by elements of \(\mathbb{R}\) as being given by \(r \cdot p=p^{r}\) where \(r \in \mathbb{R}\) and \(p \in \mathbb{R}^{+}\). With these operations, show that \(\mathbb{R}^{+}\) is a vector space over \(\mathbb{R}\).

Short answer

After checking carefully all the required properties, we can conclude that the set \(\mathbb{R}^{+}\) does indeed form a vector space over \(\mathbb{R}\) with the given operations of addition and scalar multiplication. Every single axiom for vector spaces holds for \(\mathbb{R}^{+}\) with these operations, therefore, \(\mathbb{R}^{+}\) is a vector space over \(\mathbb{R}\).

Step-by-step solution

Checking Closure under Addition and Scalar Multiplication

For \(\mathbb{R}^{+}\) to be a vector space, it should be closed under the operation of addition (multiplication in this case) and scalar multiplication. Any product or power of two positive real numbers will always be a positive real number. So, \(\mathbb{R}^{+}\) is closed under these operations.

Checking Commutativity of Addition

Next, check the commutative property of addition. Since multiplication of real numbers is always commutative, \(\forall a,b \in \mathbb{R}^{+}, a+b = b+a\) holds.

Checking Associativity of Addition

Just like the case with commutativity, multiplication of real numbers is also associative. So, \(\forall a,b,c \in \mathbb{R}^{+}, a+(b+c) = (a+b)+c\) holds true.

Checking Existence of Identity Element

The identity element for this addition operation is 1. Because for any positive real number \(a\), \(a*1 = a\). So, 1 is the additive identity in this case.

Checking Existence of Inverse Element

The additive inverse of a number is normally the number that can be added to it to get the identity element. In this case, if \(a\) is in \(\mathbb{R}^{+}\), the inverse would be \(1/a\), because \(a * 1/a = 1\), the additive identity.

Checking Distributive Laws

Finally we need to check if scalar multiplication distributes over vector addition. Given \(p,q \in \mathbb{R}^{+}\), and \(r,s \in \mathbb{R}\), we check the following:\[r \cdot (p+q) = r \cdot (p*q) = (p*q)^{r} = p^{r}*q^{r} = (r\cdot p) + (r\cdot q)\]\[ (r+s) \cdot p = p^{r+s} = p^{r}*p^{s} = (r \cdot p) + (s \cdot p)\]. So, both distributive laws hold true.

Checking Scalar Identity

Check if there is a scalar which leaves every element unchanged on scalar multiplication. Clearly, \(1 \cdot p = p^{1} = p\). So, it holds true.

Key concepts

Positive Real Numbers

Positive real numbers, denoted as \( \mathbb{R}^{+} \), are an essential concept in mathematics. They include all real numbers greater than zero, excluding zero itself. These numbers are used in various fields such as finance, physics, and statistics because they describe quantities that cannot be negative, such as length, area, or probability.

When we think about vector spaces involving positive real numbers, we need to remember that certain operations—like addition and scalar multiplication—are defined differently in this context. For instance, in \( \mathbb{R}^{+} \), the addition of two elements is defined as their multiplication. This unique definition suits applications where growth factors or scales are considered. Understanding positive real numbers in this context is crucial when discussing vector spaces since they form the backbone of this mathematical structure.

Scalar Multiplication

Scalar multiplication in the context of vector spaces over the set of positive real numbers is defined in a somewhat unconventional way. Typically, scalar multiplication involves multiplying a vector by a scalar, a real number, but here, it takes a different form.

In this setup, the operation is defined as raising a positive real number to the power of the scalar. Mathematically, it is articulated as \( r \cdot p = p^{r} \), where \( r \) is the scalar from \( \mathbb{R} \) and \( p \) is the positive real number.

This method of scalar multiplication is particularly useful because it allows us to transform a geometric progression into a series of exponents, adding a layer of exponential growth or decay to our vector space. It demonstrates how versatile mathematical definitions can be when applied appropriately to different contexts.

Associative Property

The associative property is a foundational principle in mathematics, especially when discussing operations like addition or multiplication. It states that the grouping of numbers does not affect their cumulative sum or product.

In the context of our vector space \( \mathbb{R}^{+} \), we see this property reflected through multiplication. Given this setup, for any numbers \( a, b, \) and \( c \) in \( \mathbb{R}^{+} \), it holds that \( a + (b + c) = (a + b) + c \). Translating this into addition in our defined vector space where addition is multiplication, it resembles: \( a \times (b \times c) = (a \times b) \times c \).

This property guarantees that no matter how the numbers are grouped during the operation, the result remains constant. It's an important property ensuring consistency within mathematical computations and is fundamental to maintaining a coherent structure in vector spaces.

Distributive Laws

Distributive laws are integral to understanding the structure of vector spaces. They illustrate how scalar multiplication interacts with vector addition. There are two distributive laws to consider:

• The first states that for scalars \( r \) and any vectors (which are positive real numbers in this context) \( p \) and \( q \), \( r \cdot (p+q) = r \cdot (p \times q) = (p \times q)^{r} = p^{r} \times q^{r} = (r \cdot p) + (r \cdot q) \).
• The second law addresses an addition of scalars: \( (r+s) \cdot p = p^{r+s} = p^{r} \times p^{s} = (r \cdot p) + (s \cdot p) \).

These laws ensure that the distribution of multiplication over addition—whether scalar over vectors or vectors over scalars—holds true. The consistency ensured by these properties is why distributive laws are pivotal for the vector space operations and applications.