Bases are maximal linearly independent sets

Proposition (Maximal linear independence characterization). Let $X$ be a vector space and $B\subseteq X$ . Then $B$ is a basis of $X$ if and only if:

$B$ is linearly independent, and
every strict superset $M\supsetneq B$ is linearly dependent.

Remarks

Proof sketch.

If $B$ is a basis and $x\notin B$ , then $x$ is a linear combination of elements of $B$ , so $B\cup\{x\}$ is dependent.
Conversely, if $B$ is independent and maximal and $x\notin B$ , dependence of $B\cup\{x\}$ expresses $x$ as a finite linear combination of elements of $B$ . Thus $B$ spans $X$ .