Definition:Convex Combination
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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:
- $(1): \quad \forall \alpha \in I: \lambda_\alpha > 0$
- $(2): \quad \ds \sum_{\alpha \mathop \in I} \lambda_\alpha = 1$
Examples
Arbitrary Example
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 $\mathbf c$.
Also see
- Results about convex combinations can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex combination