Span is Monotonic

Last updated Nov 1, 2022

Let $V$ be a Vector Space and let $S \subset R \subset V$. Then $\text{span}S \subset \text{span}R$.

Observe $$\text{span} R = \bigcap\limits{W \subset V : R \subset W, W \text{ is a subspace of V}} \supset R \supset S,$$ so by The Span of subset of a Span is also a subset of that Span, we have that $\text{span} S \subset \text{span} R$. $\blacksquare$

• Subspace Span