Category:Examples of Convex Combinations

This category contains examples of Convex Combination.

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:

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

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