Definition:Linear Combination

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Definition

Let $R$ be a ring.

Linear Combination of Sequence

Let $M$ be an $R$-module.

Let $\sequence {a_n} := \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a sequence of elements of $M$ of length $n$.


An element $b \in M$ is a linear combination of $\sequence {a_n}$ if and only if:

$\ds \exists \sequence {\lambda_n} \subseteq R: b = \sum_{k \mathop = 1}^n \lambda_k a_k$


Linear Combination of Subset

Let $M = \struct {G, +_G, \circ}_R$ be an $R$-module.

Let $\O \subset S \subseteq G$.

Let $b \in M$ 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 $\O$ if and only if:

$b = e_G$


Examples

Arbitrary Example

Let $\mathbf u$ and $\mathbf v$ be vectors.

Then $3 \mathbf u + 4 \mathbf v$ is a linear combination of $\mathbf u$ and $\mathbf v$.


Also see

An integer combination is also called a linear combination. The definition is compatible with the one on this page.

  • Results about linear combinations can be found here.


Sources

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