Scalar multiplication

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Not to be confused with scalar product. ‹See Tfd›

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).

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):

  • Additivity in the scalar: (c + d)v = cv + dv;
  • Additivity in the vector: c(v + w) = cv + cw;
  • Compatibility of product of scalars with scalar multiplication: (cd)v = c(dv);
  • Multiplying by 1 does not change a vector: 1v = v;
  • Multiplying by 0 gives the zero 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 of scalar multiplication is that it stretches, or contracts, vectors by a constant factor.

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]

  1. ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4. 
  2. ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6. 
  3. ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2. 
  4. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
  5. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.