One of the two operations in a vector space V, together with addition. Given any vector v in V and scalar c (from the base field), the scalar multiple cv ε‎ V may be formed. It may be that v = v is a coordinate vector, in which case the coordinates of cv are those of v multiplied by c. If v = A is a matrix, then the entries of cA are the entries of A multiplied by c. Multiplication by scalars has the following properties:

1. (i) (h+k)v = hv+kv.

2. (ii) k(v+w) = kv+kw.

3. (iii) h(kv) = (hk)v.

4. (iv) 1v = v.

Geometrically, if v is in ℝⁿ, then the scalar multiples cv comprise the line through v and the origin 0; when c > 0, the scalar mutliples comprise the half-line from 0 through v.