- Scalar multiplication
In
mathematics , scalar multiplication is one of the basic operations defining avector space inlinear algebra (or more generally, a module inabstract algebra ). Note that scalar multiplication is different fromscalar product which is aninner product between two vectors.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 denoted "cv".
Scalar multiplication obeys the following rules "(vector in
boldface )":
* Leftdistributivity : ("c" + "d")"v" = "cv" + "dv";
* Right distributivity: "c"("v" + "w") = "cv" + "cw";
*Associativity : ("cd")"v" = "c"("dv");
* Multiplying by 1 does not change a vector: 1"v" = "v";
* Multiplying by 0 gives the null vector: 0"v" = "0";
* Multiplying by -1 gives theadditive inverse : (-1)"v" = -"v".Here + isaddition 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 themultiplication 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. Ageometric 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 "K""n", 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 notcommutative , then the only change is that the order of the multiplication may be reversed from what we've written above.ee also
*
Statics
*Mechanics
*Product (mathematics)
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