Linear Combination

linear-algebra

Definition

Linear Combination

Let $V$ be a vector space over a field $F$. A linear combination of vectors $v_1,\dots ,v_n\in V$ is a sum

$$\sum _{i=1}^n{\alpha }_i{v}_i=\sum _{i=1}^n{\alpha }_i\left[\begin{array}{c}{v}_{i0}\\ v_{i1}\\ ⋮\\ v_{im}\end{array}\right]$$

where ${\alpha }_1,\dots ,{\alpha }_n\in F$ are called the coefficients.

If all ${\alpha }_i=0$, it is a trivial linear combination; otherwise it is non-trivial.