Scalar multiplicationIn mathematics, scalar multiplication is one of the basic operations defining a vector space or module in linear algebra.
Scalar multiplication obeys the following rules:
- Left distributivity: (c + d)v = cv + dv;
- Right distributivity: c(v + w) = cv + cw;
- Associativity: (cd)v = c(dv);
- Identity element: 1v = v;
- Null element: 0v = 0;
- Additive inverse element: (-1)v = -v.
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 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 from what we've written above.