Linearly independent

Vectors

Mathematical definition

A set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is linearly independent if

$$ c_1 \vec{v}_1 + c_2 \vec{v}_2 + \cdots + c_n \vec{v}_n = \vec{0} $$

has no solution besides

$$ c_1, c_2, \cdots, c_n = 0. $$

Equivalent definition

Equivalently, it can be stated that if

$$\begin{gathered} \forall i,j \in \{1,n\} \vec v_i \neq \sum \lambda\vec v_j \\ \text{(no vector in the set can be written as a linear combination of some others)} \end{gathered}$$

then set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is linearly independent