Affine Combination

linear-algebra

Definition

Affine Combination

Let ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\in V$, where $V$ is a vector space over the scalar field $F$, and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\in F$ satisfy

$$\sum _{i=1}^n{\lambda }_i=1.$$

Then the sum

$$\sum _{i=1}^n{\lambda }_i\cdot {\mathbf{v}}_i$$

is called an affine combination of ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$.

The condition ${\sum }_{i=1}^n{\lambda }_i=1$ distinguishes affine combinations from general linear combinations.

Trivial

Trivial Affine Combination

An affine combination is called trivial if exactly one coefficient is $1$ and all others are $0$, i.e. for some $k$,

$${\lambda }_k=1,{\lambda }_i=0\text{ for }i\ne k.$$

In this case, the affine combination is just ${\mathbf{v}}_k$.