Glossary

linearly dependent: The set of vectors $\{ a_{1}, ~a_{2}, \ldots, ~a_{n} \}$ are linearly dependent if there exists $\lambda_1,~\lambda_2, \ldots ~\lambda_n~\in~\mathbb{F}^{1}$, where $\lambda_{j}$ is not equal to 0 for all $n$, such that $\lambda_1 a_{1} + \lambda_2 a_2 \ldots \lambda_n a_n = 0$

linearly independent: The set of vectors $\{ a_{1}, ~a_{2}, \ldots, ~a_{n} \}$ are linearly independent if there does not exists $\lambda_1,~\lambda_2, \ldots ~\lambda_n~\in~\mathbb{F}^{1}$ such that $\lambda_1 a_{1} + \lambda_2 a_2 \ldots \lambda_n a_n = 0$ unless $\lambda_{j}$ is equal to 0 for all $n$.

A functional: takes a vector as input and outputs an element of the field the vector is defined over