Let $V\ $ be a vector space and let
$$b_1, b_2, \dots, b_n \tag{1}$$ be a system (multiset) of vectors from $V$
(multiset because the vectors $b_i$ are not required to be distinct).
Definition:
a) We will say that the system $(1)$ has rank $r \in \mathbb{N}$ if there exist $r$ linearly independent vectors from $(1)$, and every other vector from $(1)$ can be represented as a linear combination of these $r$ vectors.
b) We will say that the system $(1)$ has rank $0$ if $b_i = 0$ (the zero vector) for every $i=1,2,\dots, n$.
The rank of the system of vectors $(1)$ is denoted by $r(b_1, b_2, \dots, b_n)$.
Proposition 1: The rank of the system $(1)$ is equal to the maximal number of linearly independent vectors in the system $(1)$
Proposition 2: $r(b_1, b_2, \dots, b_n) = \dim\ \textbf{span}(b_1, b_2, \dots, b_n)$
Proposition 3: If $r(b_1, b_2, \dots, b_n)=r$ and $b$ is a vector, then $\ r(b_1, b_2, \dots, b_n, b) = r\ $ or $\ r(b_1, b_2, \dots, b_n, b) = r + 1\ $
More specifically:
A) $r(b_1, b_2, \dots, b_n, b) = r(b_1, b_2, \dots, b_n)$
if and only if $b$ is a linear combination of the vectors $b_i$
B) $r(b_1, b_2, \dots, b_n, b) = r(b_1, b_2, \dots, b_n) + 1$
if and only if $b$ is not a linear combination of the vectors $b_i$
