Scalar multiplication obeys the following rules (vector in boldface):
Here + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.
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 from what we've written above.
Publication No. WO/2010/061951 Published on June 3, Assigned to National University Okayama University for Scalar Multiplier, Scalar Multiplication Program (Japanese Inventors)
Jun 04, 2010; GENEVA, June 10 -- Yasuyuki Nogami, Yumi Sakemi and Yoshitaka Morikawa, all from Japan, have developed a scalar multiplier and...