Definition:Linear Combination
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Definition
Let $R$ be a ring.
Linear Combination of Sequence
Let $G$ be an $R$-module.
Let $\left \langle {a_n} \right \rangle := \left \langle {a_j} \right \rangle_{1 \mathop \le j \mathop \le n}$ be a sequence of elements of $G$ of length $n$.
An element $b \in G$ is a linear combination of $\left \langle {a_n} \right \rangle$ if and only if:
- $\displaystyle \exists \left \langle {\lambda_n} \right \rangle \subseteq R: b = \sum_{k \mathop = 1}^n \lambda_k a_k$
Linear Combination of Subset
Let $G$ be an $R$-module.
Let $\O \subset S \subseteq G$.
Let $b \in G$ be a linear combination of some sequence $\sequence {a_n}$ of elements of $S$.
Then $b$ is a linear combination of $S$.
Linear Combination of Empty Set
Let $G$ be an $R$-module.
$b$ is a linear combination of $\varnothing$ if and only if:
- $b = e_G$
Also see
An integer combination is also called a linear combination. The definition is compatible with the one on this page.
Sources
- 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.2$ Vector Spaces over $C$