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.
Proving Crazy Vector Space with addition and multiplication
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linear-algebra
vector-spaces
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4What 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? – 2012-03-26
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2Google "transport of structure". – 2012-03-26
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0Noone seems to figure this out. Bummer..! – 2012-04-11
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0Actually 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. – 2012-04-11