Q:

Proving Crazy Vector Space with addition and multiplication

Let $V$ be a vector space.
Let $f : V \rightarrow V$ be a bijection.  Define two new operations $+_f$ and $\cdot_f$ as follows.  If $v$ and $w$ are two vectors in $V$, $v +_f w$ is defined to be the vector $f^{-1}(f(v) + f(w))$ where $f^{-1}$ is the inverse function of  $f$. If $a$ is a scalar and $v$ is a vector in $V$, $a \cdot_f v$ is defined to be the vector $f^{-1}(af(v))$.
Prove that $V$ together with the new addition of vectors, $+_f$, and the new multiplication of vectors by scalars, $\cdot_f$, is also vector space.

A:

Replace your problem by the following: Let $X$ be a set and $V$ be a vector space, and let $f:X\to V$ be a bijection. Define operations $\oplus$ and $\otimes$ on $X$ as follows: $\ldots$
It will turn out that you have just transported the vector space structure of $V$ back to $X$, or what is the same thing: The vectors $v\in V$ have got new names $x:=f^{-1}(v)\in X$, but otherwise the operations remain unchanged.

