Linear Combination

linear-algebra

Definition

Linear Combination

Let ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\in V$, where $V$ is a vector space over the scalar field $F$, and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\in F$.

Then, the following sum is called linear combination of ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$:

$$\sum _{i=1}^n{\lambda }_i\cdot {\mathbf{v}}_i$$

where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\in F$ are called the coefficients of the linear combination.

Trivial

Trivial Linear Combination

A linear combination is called trivial if all coefficients ${\lambda }_i=0$.