properties of linear independence
Let $V$ be a vector space^{} over a field $k$. Below are some basic properties of linear independence.

1.
$S\subseteq V$ is never linearly independent^{} if $0\in S$.
Proof.
Since $1\cdot 0=0$. ∎

2.
If $S$ is linearly independent, so is any subset of $S$. As a result, if $S$ and $T$ are linearly independent, so is $S\cap T$. In addition, $\mathrm{\varnothing}$ is linearly independent, its spanning set^{} being the singleton consisting of the zero vector $0$.
Proof.
If ${r}_{1}{v}_{1}+\mathrm{\cdots}{r}_{n}{v}_{n}=0$, where ${v}_{i}\in T$, then ${v}_{i}\in S$, so ${r}_{i}=0$ for all $i=1,\mathrm{\dots},n$. ∎

3.
If ${S}_{1}\subseteq {S}_{2}\subseteq \mathrm{\cdots}$ is a chain of linearly independent subsets of $V$, so is their union.
Proof.
Let $S$ be the union. If ${r}_{1}{v}_{1}+\mathrm{\cdots}{r}_{n}{v}_{n}=0$, then ${v}_{i}\in {S}_{a(i)}$, for each $i$. Pick the largest ${S}_{a(i)}$ so that all ${v}_{i}$’s are in it. Since this set is linearly independent, ${r}_{i}=0$ for all $i$. ∎

4.
$S$ is a basis for $V$ iff $S$ is a maximal linear independent subset of $V$. Here, maximal means that any proper superset^{} of $S$ is linearly dependent.
Proof.
If $S$ is a basis for $V$, then it is linearly independent and spans $V$. If we take any vector $v\notin S$, then $v$ can be expressed as a linear combination^{} of elements in $S$, so that $S\cup \{v\}$ is no longer linearly independent, for the coefficient in front of $v$ is nonzero. Therefore, $S$ is maximal.
Conversely, suppose $S$ is a maximal linearly independent set in $V$. Let $W$ be the span of $S$. If $W\ne V$, pick an element $v\in VW$. Suppose $0={r}_{1}{v}_{1}+\mathrm{\cdots}{r}_{n}{v}_{n}+rv$, where ${v}_{i}\in S$, then $rv={r}_{1}{v}_{1}+\mathrm{\cdots}+{r}_{n}{v}_{n}$. If $r\ne 0$, then $v$ would be in the span of $S$, contradicting the assumption^{}. So $r=0$, and as a result, ${r}_{i}=0$, since $S$ is linearly independent. This shows that $S\cup \{v\}$ is linearly independent, which is impossible since $S$ is assumed to be maximal. Therefore, $W=V$. ∎
Remark. All of the properties above can be generalized to modules over rings, except the last one, where the implication^{} is only onesided: basis implying maximal linear independence.
Title  properties of linear independence 

Canonical name  PropertiesOfLinearIndependence 
Date of creation  20130322 18:05:37 
Last modified on  20130322 18:05:37 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Result 
Classification  msc 15A03 