scalarmultiplication

Scalar multiplication is an algebraic operation that combines a scalar from a field with a vector from a vector space over that field to produce another vector in the same space. It is one of the defining actions of a vector space. Formally, for a field F and a vector space V over F, there is a map F × V → V, denoted (α, v) ↦ αv, satisfying (α+β)v = αv + βv, α(u+v) = αu + αv, (αβ)v = α(βv), and 1v = v when F has a multiplicative identity 1.

In common contexts, F is the real or complex numbers, but the concept applies to any field. V may be the Euclidean space R^n, complex space C^n, a space of functions, or a module over a ring. Examples include av = (a v1, ..., a vn) in R^n, or (af)(x) = a f(x) for a function f.

Scalar multiplication interacts with other structure on V. It distributes over vector addition, and scalars can be combined with vectors to form linear combinations. Scalar multiplication is distinct from other products such as the dot product or matrix multiplication, though it often appears alongside them in linear algebra.

Applications are broad: scaling vectors in geometry and computer graphics, constructing linear combinations in solving systems and transforming spaces, and, in more advanced settings, serving as the action of scalars in modules or in cryptographic schemes that involve scalar multiplication on algebraic groups.