# 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 common geometrical contexts, 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

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

*c*.

**v**### Properties

Scalar multiplication obeys the following rules *(vector in boldface)*:

- Additivity in the scalar: (
*c*+*d*)=**v***c*+**v***d*;**v** - Additivity in the vector:
*c*(+**v**) =**w***c*+**v***c*;**w** - Compatibility of product of scalars with scalar multiplication: (
*cd*)=**v***c*(*d*);**v** - Multiplying by 1 does not change a vector: 1
=**v**;**v** - Multiplying by 0 gives the zero vector: 0
=**v**;**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

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 *K*^{n}, 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* *c v* and

*right scalar multiplication*

*may be defined.*

**v**c## Scalar multiplication of matrices

The **left scalar multiplication** of a matrix **A** with a scalar *λ* gives another matrix *λ***A** of the same size as **A**. The entries of *λ***A** are defined by

explicitly:

Similarly, the **right scalar multiplication** of a matrix **A** with a scalar *λ* is defined to be

explicitly:

When the underlying ring is commutative, for example, the real or complex number field, these two multiplications are the same, and are simply called *scalar multiplication*. However, for matrices over a more general ring that are *not* commutative, such as the quaternions, they may not be equal.

For a real scalar and matrix:

For quaternion scalars and matrices:

where *i*, *j*, *k* are the quaternion units. The non-commutativity of quaternion multiplication prevents the transition of changing *ij* = +*k* to *ji* = −*k*.

## See also

## References

- 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.