Definition:Convex Combination

Definition

Let $K$ be a field.

Let $V$ be a vector space over $K$.

Let $\family {\mathbf v_\alpha}_{\alpha \mathop \in I} \subseteq V$ be a family of elements of $V$ indexed by an indexing set $I$.

Let $\ds \sum_{\alpha \mathop \in I} \lambda_\alpha \mathbf v_\alpha$ be a linear combination of $\family {\mathbf v_\alpha}$.

Then $\ds \sum_{\alpha \mathop \in I} \lambda_\alpha \mathbf v_\alpha$ is a convex combination of $\family {\mathbf v_\alpha}$ if and only if:

$(1): \quad \forall \alpha \in I: \lambda_\alpha > 0$

$(2): \quad \ds \sum_{\alpha \mathop \in I} \lambda_\alpha = 1$

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors of a vector space $V$.

Then:

$0.1 \mathbf a + 0.3 \mathbf b + 0.6 \mathbf c$ is a convex combination of $\mathbf a$, $\mathbf b$ and $\mathbf c$.