Proving that something is a vector space using a map between vector spaces. I'm doing a problem set, and I came about a problem.
Let $V$ be the set of positive reals.  Define $ x “+” y = x y$, and for $ \alpha \in \mathbf{R} $, define $ “\alpha x” = x^a $. Is $V$ a vector space with these unusual operations of addition and scalar multiplication? 
This is actually seems like a very easy problem. It's pretty clear that $V$ is a vector space. Now, I could go through the definition of a vector space and show that $ V $ satisfies all of the characteristics, but that wouldn't be very interesting or fun.
I was hoping I could come up with a more interesting proof. I noticed pretty quickly that you could use logarithms to show a relation to $ \mathbf{R} $ under the usual operations of addition and scalar multiplication, which we proved in class is a vector field and thus I wouldn't have to prove it again. The trouble is that I'm not sure if and how this can be rigorously proven.
This is how I begun
Consider the logarithm, a bijection from $\mathbf{R}^+$ to $\mathbf{R}$. Note that for $x,y \in \mathbf{R}^+ $, and $ \alpha \in \mathbf{R} $, it follows that $ \log x + \log y = \log \left( x “+” y \right) $ and $ \alpha \log x = \log \left( “\alpha x” \right) $.
I don't know how to proceed.
PS: My professor used the notation $ x “+” y $ and $ “\alpha x” $ for arbitrary vector sums and scalar multiplication when he wanted to distinguish them from standard sums and multiplication. Is there a better way do distinguish them?
 A: To answer your second question first: $\oplus$ and $\odot$ are sometimes used to denote "nonstandard" addition and multiplication.
A structure-preserving map between vector spaces is a linear map or vector-space homomorphism $f:V\to W$ satisfying
$$f(v+w) = f(v)\oplus f(w)$$
$$f(\alpha v) = \alpha \odot f(v)$$
and you should be able to prove that the image of a vector space $V$ under a vector-space homomorphism $f$ is also a vector space (with the same field). 
You do this by checking the vector space axioms: for example, let $u,v,w$ be vectors in $W$. Then there exist vectors $\bar u$, $\bar v$, $\bar w$ in $V$ with $f(\bar u) = u$, etc. Therefore
\begin{align*}
u \oplus (v \oplus w) &= u \oplus (f(\bar v) \oplus f(\bar w))\\
 &= u \oplus f(\bar v + \bar w)\\
 &= f(\bar u) \oplus f(\bar v + \bar w)\\
 &= f(\bar u + (\bar v + \bar w))\\
&= f((\bar u + \bar v) + \bar w)\\
&= f(\bar u + \bar v) \oplus f(\bar w)\\
&= (f(\bar u) \oplus f(\bar v)) \oplus w\\
&= (u \oplus v) \oplus w,
\end{align*}
and therefore $\oplus$ is associative. You now need to similarly prove the other axioms.
As a practical matter, proving this theorem about $f$ may be more work than proving the vector space axioms directly (but on the upside, will let you apply the theorem to later problems).
