Prove that $V$ is a vector space over $R$

Question

Let $V=(0,\infty)$ and let $x,y \in V , a\in R$

Define $x+y = xy\ \ $ and $\ \ ax=x^a$

Prove the $V$ is a vector space over $R$

My input:

I am trying to prove all the axioms.

$1. $ let $x_1\ \ $and $ \ y_1 \in V$

and $ x_1 + y_1=x_1y_1 $

$ y_1 + x_1=y_1x_1 $

Now I am stuck it this very first step. How can I tell that $\ \ y_1x_1=x_1y_1$ ? I am learning basics so please give me advise or something before getting deep into linear algebra.

Answer 1

Guide:

You should use properties inherited from (positive) real numbers.

For example:

Multiplication of real numbers are commutative, hence we have $y_1x_1=x_1y_1$ since $x_1, y_1 \in \mathbb{R}$. Hence addition that you have defined is commutative.