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### Rank of a system of vectors

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 $b_i$