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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, scalar multiplication is one of the basic operations defining a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
(or more generally, a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
). In common geometrical contexts, scalar multiplication of a real
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
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 Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number w ...
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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 ''k'' in ''K'' and v in ''V'' is denoted ''k''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: 1v = v; * Multiplying by 0 gives the zero vector: 0v = 0; * Multiplying by −1 gives the
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
: (−1)v = −v. Here, + is
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
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 Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
operation in the field.


Interpretation

Scalar multiplication may be viewed as an external
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
or as an action of the field on the vector space. A
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
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. 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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
and ''V'' is a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
over ''K''. ''K'' can even be a rig, but then there is no additive inverse. If ''K'' is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, the distinct operations ''left scalar multiplication'' ''c''v and ''right scalar multiplication'' v''c'' may be defined.


Scalar multiplication of matrices

The left scalar multiplication of a matrix with a scalar gives another matrix of the same size as . It is denoted by , whose entries of are defined by : (\lambda \mathbf)_ = \lambda\left(\mathbf\right)_\,, explicitly: : \lambda \mathbf = \lambda \begin A_ & A_ & \cdots & A_ \\ A_ & A_ & \cdots & A_ \\ \vdots & \vdots & \ddots & \vdots \\ A_ & A_ & \cdots & A_ \\ \end = \begin \lambda A_ & \lambda A_ & \cdots & \lambda A_ \\ \lambda A_ & \lambda A_ & \cdots & \lambda A_ \\ \vdots & \vdots & \ddots & \vdots \\ \lambda A_ & \lambda A_ & \cdots & \lambda A_ \\ \end\,. Similarly, even though there is no widely-accepted definition, the right scalar multiplication of a matrix with a scalar could be defined to be : (\mathbf\lambda)_ = \left(\mathbf\right)_ \lambda\,, explicitly: : \mathbf\lambda = \begin A_ & A_ & \cdots & A_ \\ A_ & A_ & \cdots & A_ \\ \vdots & \vdots & \ddots & \vdots \\ A_ & A_ & \cdots & A_ \\ \end\lambda = \begin A_ \lambda & A_ \lambda & \cdots & A_ \lambda \\ A_ \lambda & A_ \lambda & \cdots & A_ \lambda \\ \vdots & \vdots & \ddots & \vdots \\ A_ \lambda & A_ \lambda & \cdots & A_ \lambda \\ \end\,. When the entries of the matrix and the scalars are from the same commutative field, for example, the real number field or the complex number field, these two multiplications are the same, and can be simply called ''scalar multiplication''. For matrices over a more general field that is ''not'' commutative, they may not be equal. For a real scalar and matrix: : \lambda = 2, \quad \mathbf =\begin a & b \\ c & d \\ \end : 2 \mathbf = 2 \begin a & b \\ c & d \\ \end = \begin 2 \!\cdot\! a & 2 \!\cdot\! b \\ 2 \!\cdot\! c & 2 \!\cdot\! d \\ \end = \begin a \!\cdot\! 2 & b \!\cdot\! 2 \\ c \!\cdot\! 2 & d \!\cdot\! 2 \\ \end = \begin a & b \\ c & d \\ \end2= \mathbf2. For quaternion scalars and matrices: : \lambda = i, \quad \mathbf = \begin i & 0 \\ 0 & j \\ \end : i\begin i & 0 \\ 0 & j \\ \end = \begin i^2 & 0 \\ 0 & ij \\ \end = \begin -1 & 0 \\ 0 & k \\ \end \ne \begin -1 & 0 \\ 0 & -k \\ \end = \begin i^2 & 0 \\ 0 & ji \\ \end = \begin i & 0 \\ 0 & j \\ \endi\,, where are the quaternion units. The non-commutativity of quaternion multiplication prevents the transition of changing to .


See also

*
Dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
* Matrix multiplication *
Multiplication of vectors In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: * Dot product In mathematics, the dot product or scalar productThe term ''scalar product'' ...
*
Product (mathematics) In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\cd ...


References

Linear algebra Abstract algebra Multiplication {{Improve categories, date=May 2021