Linear Combination
A linear combination of the vectors $$\left\{ v _ 1, \ldots, v _ m \right\}$$ of vectors in $$V$$ $a _ 1 v _ 1 + \ldots + a_m v_m$ where $$a_1, \ldots, a_m \in \mathbb{F}$$
Span
The set of all linear combinations of the vectors $$\left\{ v _ 1, \ldots, v _ m \right\} \subseteq V$$ called the span of $$\left\{ v_1, \ldots, v_m \right\}$$ and we define the notation: $\operatorname{ span } \left( \left\{ v_1, ..., v_m \right\} \right) := \left\{ a_1 v_1 + \ldots + a_m v_m: a_1, ..., a_m \in \mathbb { F } \right\}$ we define $$\operatorname{ span } \left( \emptyset \right) := \left\{ 0 \right\}$$
Linearly Independent
A set of vectors $$\left\{ v _ 1, \ldots, v _ m \right\} \subseteq V$$ is said to be linearly independent if the only solution to $$\sum _ { i = 1 } ^ m a_i v_i = 0$$ is $$\forall i \in \left\{ 1, ..., m \right\}, a_i = 0$$. We define $$\emptyset$$ to be linearly independent.
Linearly Independent iff Unique Representation
$$\left\{ v_1, \ldots, v_m \right\}$$ is linearly independent iff each vector in the set $$\left\{ v_1, \ldots, v_m \right\}$$ has only one representation as a linear combination of $$\left\{ v_1, \ldots, v_m \right\}$$
Linearly Dependent
$$\left\{ v_1, \ldots, v_m \right\} \subseteq V$$ is said to be linearly dependent if it is not linearly independent
Linearly Depedent iff Zero has a Non-Trivial Representation
$$\left\{ v_1, \ldots, v_m \right\} \subseteq V$$ is linearly dependent iff there exists $$a_1, \ldots, a_m \in \mathbb { F }$$ and $$i \in \left[ 1,m \right]$$ where $$a_i \neq 0$$ such that $$\sum _ { i = 1 } ^ m a_i v_i = 0$$