Bases are maximal linearly independent sets

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

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

Context. This proposition identifies bases with “maximal independent sets”: you cannot add any new vector to $B$ without creating a nontrivial linear dependence.

Proof sketch.

• ($\Rightarrow$) If $B$ is a basis and $x\notin B$, then $x$ is a linear combination of elements of $B$, so $\{x\}\cup B$ is dependent; hence any strict superset of $B$ is dependent.
• ($\Leftarrow$) If $B$ is independent and maximal, then for any $x\in X\setminus B$, the set $B\cup\{x\}$ is dependent, so one can solve for $x$ as a linear combination of finitely many elements of $B$; thus $B$ spans $X$.