🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

A linear combination of the vectors $\{v_1,\dots ,v_m\}$ of vectors in a vector space $V$ is: $${\displaystyle a_1{v}_1+\dots +a_m{v}_m}$$ 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: $${\displaystyle span\left(\{v_1,...,v_m\}\right):=\{a_1{v}_1+\dots +a_m{v}_m:a_1,...,a_m\in \mathbb{F}\}}$$ we define $span(\varnothing ):=\left\{0\right\}$

A set of vectors $\{v_1,\dots ,v_m\}\subseteq V$ is said to be linearly independent if the only solution to $${\displaystyle {\sum }_{i=1}^m{a}_i{v}_i=0}$$ is $\forall i\in \{1,...,m\},a_i=0$. We define $\varnothing$ to be linearly independent.

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

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