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 \( \sum _ { i = 1 } ^ m a_i v_i = 0 \)