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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.

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What have you tried? Your problem is to prove that $V$ is a vector space under the new operation. What would you need to show in order to have proved this? –  MJD Mar 26 '12 at 17:42
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Google "transport of structure". –  Bill Dubuque Mar 26 '12 at 17:48
    
Noone seems to figure this out. Bummer..! –  user28873 Apr 11 '12 at 22:07
    
Actually Bill's comment is a nearly complete answer. Furthermore this is really nothing more than a simple exercise in verification that the definition of "vector space" holds with the new operations. –  Asaf Karagila Apr 11 '12 at 22:48

1 Answer 1

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.

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