I have the following definition: let V be a vector space over

a division ring D. A mapping a of V into V is called a linear trans-

formation of V if it has the followiwng two properties:

(x+y)a = xa+ya for x,y \(\displaystyle \in\) V,

(x\(\displaystyle \alpha\))a = (xa)\(\displaystyle \alpha\) for x \(\displaystyle \in\) V, \(\displaystyle \alpha\) D

And here I find an odd thing. If a is the mapping multiplication by

a scalar (that is, by an element of D), then a is not in general a linear trans-

formation of V according to the definition, because D needs not be

commutative. Any hint will be welcome.