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)$
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?