More specifically, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is cv.
Scalar multiplication obeys the following rules:
Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space.
As a special case, V may be taken to be K itself and scalar multiplciation may then be taken to be simply the multiplciation in the field. When V is K^{n}, then scalar multiplication is defined componentwise[?].
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 from what we've written above.
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