A linear combination of the vectors $$ \{v_1,\dots <!--\; \dots \; -->,v_m\}$$ of vectors in $$ V$$ $\[ a_1{v}_1+\dots <!--\; \dots \; -->+a_m{v}_m \]$ where $$ a_1,\dots <!--\; \dots \; -->,a_m\in <!--\; \in \; -->\mathbb{F}$$

The set of all linear combinations of the vectors $$ \{v_1,\dots <!--\; \dots \; -->,v_m\}\subseteq <!--\; \subseteq \; -->V$$ called the span of $$ \{v_1,\dots <!--\; \dots \; -->,v_m\}$$ and we define the notation: $\[ \mathrm{span}<!--\; \; -->\left(\{v_1,...,v_m\}\right):=\{a_1{v}_1+\dots <!--\; \dots \; -->+a_m{v}_m:a_1,...,a_m\in <!--\; \in \; -->\mathbb{F}\} \]$ we define $$ \mathrm{span}<!--\; \; -->\left(\mathrm{\varnothing <!--\; \varnothing \; -->}\right):=\left\{0\right\}$$

A set of vectors $$ \{v_1,\dots <!--\; \dots \; -->,v_m\}\subseteq <!--\; \subseteq \; -->V$$ is said to be linearly independent if the only solution to $$ \underset{i=1}{\overset{m}{\sum <!--\; \sum \; -->}}{a}_i{v}_i=0$$ is $$ \mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,...,m\},a_i=0$$. We define $$ \mathrm{\varnothing <!--\; \varnothing \; -->}$$ to be linearly independent.

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