Scalar

Intuition

A scalar is a number by which one can multiply a vector^FR. Most of the time, scalars are simply real or complex numbers.

A vector $v$ multiplied by the scalar $2$.

Formal definition

Let $E$ be a vector space over a field $K$. The elements of $K$ are called the scalars of $E$.
Scalar multiplication is the map
$$K \times E \longrightarrow E, \quad(\lambda, v) \longmapsto \lambda v.$$

Remarks

• More generally, if $M$ is a module over a ring $A$, the elements of $A$ may also be called scalars.