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Problem 50: Determine whether $\R^+$, is a vector space $x + y = xy$,
$c\cdot x = x^c$.

Problem 50: On $\R^+$, the set of positive real numbers, define the operations of addition and scalar multiplication as follows:
$x + y = xy$,
$c\cdot x = x^c$.
Note that the multiplication and exponentiation appearing on the right side of these formulas refer to the ordinary operations on real numbers.
Determine whether $\R^+$, together with these algebraic operations,is a vector space.

Solution:
The Axioms of a Vector Space
The following properties must hold for all $ u,v,w \in \V$ and $a,b \in \R$:
Closure Properties
$(A1) u+v \in \V$.
$(A2) av \in \V$.
Properties of Addition
$(A3) u+v=v+u$.
$(A4) u+(v+w)=(u+v)+w$.
$(A5)$ There is an element $\mathbf{0} \in \V$ such that $\mathbf{0}+v=v$ for all $v \in \V$.
$(A6)$ Given an element $v \in \V$, there is an element $−v \in \V$ such that $v+(−v)=\mathbf{0}$.
Properties of Scalar Multiplication
$(A7) a(bv)=(ab)v$.
$(A8) a(u+v)=au+av$.
$(A9) (a+b)v=av+bv$.
$(A10) 1v=v$ for all $v \in \V$.

Prove A1:
Let $x,y \in \R^+$
\begin{align*} x+y = xy \in \R^+ \end{align*}

Prove A2:
Let $r \in \R$ and $x \in \R^+$
\begin{align*} rx = x^r \in \R^+ \end{align*}

Prove A3:
Let $x,y \in \R^+$
\begin{align*} x+y = xy = yx = y+x \end{align*}

Prove A4:
Let $x,y,z \in \R^+$
\begin{align*} (x+y)+z = (xy)+z\\ = (xy)z \\ = xyz \\ = x(yz) \\ = x(y+z) \\ = x+(y+z). \end{align*}

Prove A5:
Let $x\in \R^+$ and $1 \in \R^+$
zero vector in this set is the real number $1$. \begin{align*} 1+x = 1x\\ = x\\ = x1\\ = x+1 \end{align*}

Prove A6:
Let $x \in \R^+$ then additive inverese is $\frac{1}{x} \in \R^+$
\begin{align*} x+\frac{1}{x} = x\frac{1}{x} \\ = 1 \\ = \frac{1}{x}x\\ = \frac{1}{x}+x \end{align*}

Prove A7:
Let $x \in \R^+$ and $r,s \in \R$
\begin{align*} (rs)x = x^{rs} \\ = x^{sr} \\ = (x^s)^r \\ = r\cdot x^s \\ = r\cdot (s\cdot x) = r(sx) \end{align*}

Prove A8:
Let $x,y \in \R^+$ and $r \in \R$
\begin{align*} r(x+y) = (x+y)^r\\ = (xy)^r\\ = x^r+y^r\\ = rx+ry. \end{align*}

Prove A9:
Let $x\in \R^+$ and $r,s \in \R$
\begin{align*} (r+s)x = x^(r+s)\\ = x^rx^s\\ = x^r+x^s\\ = rx+sx \end{align*}

Prove A10:
Let $x\in \R^+$ and $1 \in \R$
\begin{align*} 1\cdot x =x^1\\ = x \end{align*}

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