Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra[1][2][3] (or more generally, a module in abstract algebra[4][5]). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and must be distinguished from inner product of two vectors (where the product is a scalar).
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Definition[edit]
In general, 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.
Properties[edit]
Scalar multiplication obeys the following rules (vector in boldface):
- Left distributivity: (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: 1v = v;
- Multiplying by 0 gives the null vector: 0v = 0;
- Multiplying by −1 gives the additive inverse: (−1)v = −v.
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.
Interpretation[edit]
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, scalar multiplication is equivalent to multiplication of each component with the scalar, and may be defined as such.
The same idea applies 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, the distinct operations left scalar multiplication cv and right scalar multiplication vc may be defined.
See also[edit]
References[edit]
- ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4.
- ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.
- ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2.
- ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
- ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
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