Linear Combination

A linear combination of the vectors $$ \{{v}_{1},\dots ,{v}_{m}\}$$ of vectors in $$ V$$
$$${a}_{1}{v}_{1}+\dots +{a}_{m}{v}_{m}$$$
where $$ {a}_{1},\dots ,{a}_{m}\in \mathbb{F}$$

Span

The set of all linear combinations of the vectors $$ \{{v}_{1},\dots ,{v}_{m}\}\subseteq V$$ called the span of $$ \{{v}_{1},\dots ,{v}_{m}\}$$ and we define the notation:
$$$\mathrm{span}\left(\{{v}_{1},...,{v}_{m}\}\right):=\{{a}_{1}{v}_{1}+\dots +{a}_{m}{v}_{m}:{a}_{1},...,{a}_{m}\in \mathbb{F}\}$$$
we define $$ \mathrm{span}\left(\mathrm{\varnothing}\right):=\left\{0\right\}$$

Linearly Independent

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

Linearly Independent iff Unique Representation

$$ \{{v}_{1},\dots ,{v}_{m}\}$$ is linearly independent iff each vector in the set $$ \{{v}_{1},\dots ,{v}_{m}\}$$ has only one representation as a linear combination of $$ \{{v}_{1},\dots ,{v}_{m}\}$$

Linearly Dependent

$$ \{{v}_{1},\dots ,{v}_{m}\}\subseteq V$$ is said to be linearly dependent if it is not linearly independent

Linearly Depedent iff Zero has a Non-Trivial Representation

$$ \{{v}_{1},\dots ,{v}_{m}\}\subseteq V$$ is linearly dependent iff there exists $$ {a}_{1},\dots ,{a}_{m}\in \mathbb{F}$$ and $$ i\in [1,m]$$ where $$ {a}_{i}\ne 0$$ such that $$ \sum _{i=1}^{m}{a}_{i}{v}_{i}=0$$