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Definition-A

Let $V$ be a vector space over a field $F$, then the mapping $T: V \rightarrow V $ is called linear transformation if the following conditions are satisfied. (i) $T(x+y)=T(x)+T(y)$ (ii) $ T(\alpha x)= \alpha T(x)$

Definition-B

Let $V$ be a vector space over a field $F$, then mapping $T: V \rightarrow V $ is called linear transformation if $T(\alpha x+\beta y)=\alpha T(x)+\beta T(y)$

Question is some authors use definition-A and others use definition-B. Whether both of them are equivalent. Clearly Definition-A implies definition-B. How to prove definition-B implies definition-A?

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To get $(i)$ put $\alpha=\beta=1$ and to get $(ii)$ put $\alpha=1, \beta=0.$

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  • $\begingroup$ Assume that V is semigroup with respect to addition and F is semiring without multiplicative identity 1. Then is it possible? $\endgroup$ – Petchimuthu Subramanian Nov 9 '16 at 13:25
  • $\begingroup$ Well...Without $1$ my proof becomes incorrect $\endgroup$ – Leox Nov 9 '16 at 15:34
  • $\begingroup$ Dear Mr. Leox Thank for your proof and comments . So A implies B always true, but another one need not be true in general am I right ? $\endgroup$ – Petchimuthu Subramanian Nov 10 '16 at 10:58
  • $\begingroup$ I think that for semigroups you may use the definition B and forget the definition A $\endgroup$ – Leox Nov 10 '16 at 21:36
  • $\begingroup$ Yes it is good idea thank you for your suggestion. $\endgroup$ – Petchimuthu Subramanian Nov 11 '16 at 10:52
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$T(\alpha x+ \beta y) = T(\alpha x) + T (\beta y) = \alpha T(x) + \beta T(y)$.

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