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).

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.

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.

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.^[6]

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.

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, whose entries of λA are defined by

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

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.

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