Linear Combinations

Linear combination

Definition 1(Definition).

Let $V$ be a vector space, and let ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3,\cdots {\mathbf{v}}_n\in V$ be a collection of vectors. A linear combination of these vectors is a sum of the form

$${\alpha }_1{\mathbf{v}}_1+{\alpha }_2{\mathbf{v}}_2+\dots +{\alpha }_p{\mathbf{v}}_p=\sum _{k=1}^p{\alpha }_k{\mathbf{v}}_k.$$

A linear combination is called trivial if all the coefficients are zero.

Span

Definition 2(Definition).

Given a system of vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_r\in V$, its linear span is the collection of all vectors $\mathbf{v}\in V$ that can be represented as a linear combination of the system.