In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). 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. 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.

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

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.

The space of vectors may be considered a coordinate space where elements are associated with a list of elements from K. The units of the field form a group K ^× and the scalar-vector multiplication is a group action on the coordinate space by K ^×. The zero of the field acts on the coordinate space to collapse it to the zero vector.

When K is the field of real numbers there is a geometric interpretation of scalar multiplication: 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 Kⁿ, scalar multiplication is equivalent to multiplication of each component with the scalar, and may be defined as such.