Definition:Linear Combination

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 $\varnothing \subset S \subseteq G$.

Let $b \in G$ be a linear combination of some sequence $\left \langle {a_n} \right \rangle$ 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

• Definition:Linear Span
• Definition:Linear Transformation

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