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, $\varnothing$ is linearly independent, its spanning set being the singleton consisting of the zero vector $0$ .
   Proof. If $r_1v_1+\cdots r_nv_n=0$ , where $v_i\in T$ , then $v_i\in S$ , so $r_i=0$ for all $i=1,\ldots, n$ .
3. If $S_1\subseteq S_2\subseteq \cdots$ is a chain of linearly independent subsets of $V$ , so is their union.
   Proof. Let $S$ be the union. If $r_1v_1+\cdots r_nv_n=0$ , 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 $S\cup \lbrace v\rbrace$ is no longer linearly independent, for the coefficient in front of $v$ is non-zero. 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 $v\in V-W$ . Suppose $0=r_1v_1+\cdots r_nv_n+rv$ , where $v_i\in S$ , then $-rv = r_1v_1+\cdots +r_nv_n$ . 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 $S\cup \lbrace v\rbrace$ 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 one-sided: basis implying maximal linear independence.