Is R^+ a Vector Space with Non-Standard Operations?

[mpm]

I have a homework problem that I can't figure out and there is nothing in the book that helps me out. I was hoping someone could shed some light.

Let R^+ denote the set of postive real numbers. Define the operation of scalar muplication, denoted * (dot) by,

a*x = x^a

for each X (episilon) R^+ and for any real number a. Define the operation of addition, denoted +, by

x + y = x * y for all x, y (Epsilon)R^+

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

- 3 * 1/2 = (1/2)^-3 = 8

and the sume of 2 and 5 is given by

2 + 5 = 2 * 5 = 10

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

The plus should be a plus with a circle around it but I couldn't figure out how to put it in there. I am also not sure how to make the epsilon either.

Any help would be greatly appreciated.

mpm

 

[AKG]

This is an epsilon ([itex]\epsilon[/itex]), you're looking for a different symbol, the "is a member or element of" relation ([itex]\in[/itex]). So you have:

[tex](\mathbb{R}^+, \oplus, \otimes)[/tex]

with the following definitions, for all x, y in R⁺ and all scalars (reals) [itex]\lambda[/itex]:

[tex]x \oplus y = x \times y[/tex]

[tex]\lambda \otimes x = x^{\lambda}[/tex]

Do you know the definition of a vector space? Basically, all you have to do is check that the operations are well-defined, and then show that they satisfy all the properties (like commutativity of addition, associativity of scalar multiplication, existence of identities, etc.).

 

[quicknote]

I understand your frustration with not being able to solve a homework problem. However, I am unable to provide a direct response to your question as it is important for you to work through the problem and understand the concepts on your own.

However, I can offer some guidance and suggestions to help you solve this problem. Firstly, it is important to understand the definitions of a vector space and its operations. A vector space is a set of elements that can be added and multiplied by scalars to produce new elements within the set. In this case, R^+ is the set of positive real numbers and the operations defined are scalar multiplication and addition.

To prove whether R^+ is a vector space with these operations, you need to show that it satisfies the 10 axioms (properties) of a vector space. These axioms include properties such as closure under addition and scalar multiplication, associativity, commutativity, and distributivity. You can refer to your textbook or other resources for a detailed explanation of these axioms.

To start, you can check if R^+ satisfies the closure property under addition and scalar multiplication. This means that when you add or multiply two elements from R^+, the result should also be an element of R^+. In this case, for any x and y in R^+, x + y and a*x should also be in R^+.

Next, you can check the other axioms to see if they hold true for R^+. If any of the axioms do not hold true, then R^+ is not a vector space with the given operations. If all 10 axioms are satisfied, then you can conclude that R^+ is a vector space with these operations.

I hope this helps guide you in solving the problem. Remember, as a scientist, it is important to approach problems critically and logically, using your knowledge and resources to arrive at a solution. Good luck!