# Scalar multiplication

Scalar multiplication of a vector by a factor of 3 stretches the vector out.
The scalar multiplications −a and 2a of a vector a

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 cv.

### Properties

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

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.

## Scalar multiplication of matrices

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

${\displaystyle (\lambda \mathbf )_=\lambda \left(\mathbf \right)_\,,}$

explicitly:

${\displaystyle \lambda \mathbf =\lambda {\beginA_&A_&\cdots &A_\\A_&A_&\cdots &A_\\\vdots &\vdots &\ddots &\vdots \\A_&A_&\cdots &A_\\\end}={\begin\lambda A_&\lambda A_&\cdots &\lambda A_\\\lambda A_&\lambda A_&\cdots &\lambda A_\\\vdots &\vdots &\ddots &\vdots \\\lambda A_&\lambda A_&\cdots &\lambda A_\\\end}\,.}$

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

${\displaystyle (\mathbf \lambda )_=\left(\mathbf \right)_\lambda \,,}$

explicitly:

${\displaystyle \mathbf \lambda ={\beginA_&A_&\cdots &A_\\A_&A_&\cdots &A_\\\vdots &\vdots &\ddots &\vdots \\A_&A_&\cdots &A_\\\end}\lambda ={\beginA_\lambda &A_\lambda &\cdots &A_\lambda \\A_\lambda &A_\lambda &\cdots &A_\lambda \\\vdots &\vdots &\ddots &\vdots \\A_\lambda &A_\lambda &\cdots &A_\lambda \\\end}\,.}$

When the 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:

${\displaystyle \lambda =2,\quad \mathbf ={\begina&b\\c&d\\\end}}$
${\displaystyle 2\mathbf =2{\begina&b\\c&d\\\end}={\begin2\!\cdot \!a&2\!\cdot \!b\\2\!\cdot \!c&2\!\cdot \!d\\\end}={\begina\!\cdot \!2&b\!\cdot \!2\\c\!\cdot \!2&d\!\cdot \!2\\\end}={\begina&b\\c&d\\\end}2=\mathbf 2.}$

For quaternion scalars and matrices:

${\displaystyle \lambda =i,\quad \mathbf ={\begini&0\\0&j\\\end}}$
${\displaystyle i{\begini&0\\0&j\\\end}={\begini^&0\\0&ij\\\end}={\begin-1&0\\0&k\\\end}\neq {\begin-1&0\\0&-k\\\end}={\begini^&0\\0&ji\\\end}={\begini&0\\0&j\\\end}i\,,}$

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