Interpretation
Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.
As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field.
When V is Kn, then scalar multiplication is defined component-wise.
The same idea goes through with no change if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, then the only change is that the order of the multiplication may be reversed, resulting in the distinct operations left scalar multiplication cv and right scalar multiplication vc.
Read more about this topic: Scalar Multiplication