# linear transfomation

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?

## 2 Answers

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

• Assume that V is semigroup with respect to addition and F is semiring without multiplicative identity 1. Then is it possible? – Petchimuthu Subramanian Nov 9 '16 at 13:25
• Well...Without $1$ my proof becomes incorrect – Leox Nov 9 '16 at 15:34
• 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 ? – Petchimuthu Subramanian Nov 10 '16 at 10:58
• I think that for semigroups you may use the definition B and forget the definition A – Leox Nov 10 '16 at 21:36
• Yes it is good idea thank you for your suggestion. – Petchimuthu Subramanian Nov 11 '16 at 10:52

$T(\alpha x+ \beta y) = T(\alpha x) + T (\beta y) = \alpha T(x) + \beta T(y)$.