Category:Definitions/Convex Combinations
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This category contains definitions related to Convex Combinations.
Related results can be found in Category:Convex Combinations.
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:
- $(1): \quad \forall \alpha \in I: \lambda_\alpha > 0$
- $(2): \quad \ds \sum_{\alpha \mathop \in I} \lambda_\alpha = 1$
Pages in category "Definitions/Convex Combinations"
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