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 is to 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 k in K and v in V is denoted kv.[6]
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 result, it produces a vector in the same or opposite direction of the original vector but of a different length.[7]
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.
Scalar multiplication of matrices[edit]
The left scalar multiplication of a matrix A with a scalar λ gives another matrix of the same size as A. It is denoted by λA,[6] whose entries of λA 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[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.
- ^ a b "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-06.
- ^ Weisstein, Eric W. "Scalar Multiplication". mathworld.wolfram.com. Retrieved 2020-09-06.