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A linear combination of the vectors $\{v_1,\dots ,v_m\}$ of vectors in a vector space $V$ is: $$a_1{v}_1+\dots +a_m{v}_m$$ where $a_1,\dots ,a_m\in 𝔽$

The set of all linear combinations of the vectors $\{v_1,\dots ,v_m\}\subseteq V$ (where $V$ is a vector space) called the span of $\{v_1,\dots ,v_m\}$ and we define the notation: $$\mathrm{span}\left(\{v_1,...,v_m\}\right):=\{a_1{v}_1+\dots +a_m{v}_m:a_1,...,a_m\in 𝔽\}$$ we define $\mathrm{span}\left(\varnothing \right):=\left\{0\right\}$

Let $V$ be a vector space over a field $F$. A set of vectors $\{v_1,\dots ,v_m\}\subseteq V$ is linearly independent if, for scalars $a_1,\dots ,a_m\in F$, the only solution to $$\sum _{i=1}^m{a}_i{v}_i=0$$ is $a_1=\cdots =a_m=0$. We define $\varnothing$ to be linearly independent.

A set $S\subseteq V$ is said to be generally linearly independent if for every every finite subset $F\subseteq S$, $F$ is linearly indepenent

Suppose that $V$ is a vector space, then we say that $S$ is a spanning set of $V$ if $\mathrm{span}\left(S\right)=V$

$\{v_1,\dots ,v_m\}\subseteq V$ is said to be linearly dependent if it is not linearly independent