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. 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).
Not to be confused with scalar product.Interpretation[edit]
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.[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.