Elementary Properties of Linear Independence

Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$, let $n$ be a natural number with the order relations $<$ and $\le$, and let $v\in V^{n}$ be an $n$-tuple in $V$. For $j\in[n]$, with $[j]$ the initial segment determined by $j$, write $v|_{[j]}\in V^{j}$ for the restriction of $v$ to $[j]$. Then the following hold.

1. (Restriction) If $v$ is linearly independent and $j\in[n]$, then $v|_{[j]}$ is linearly independent.

2. (Predecessors) If $v$ is linearly independent, then $v_{1}\ne 0_{V}$, and for every $m\in\mathbb{N}$ with $m+1\in[n]$ the component $v_{m+1}$ does not lie in the span of $v|_{[m]}$.

3. (Dependence) Suppose $n=p+1$ for some $p\in\mathbb{N}$, and that $v$ is not linearly independent. Then there is $j\in[n]$ with

$$v_{j}\in\operatorname{span}\bigl(v^{(j)}\bigr),$$

where $v^{(j)}\in V^{p}$ is obtained from $v$ by omitting the $j$-th component, as in Extraction of a Summand from a Finite Sum of Vectors.