The Span of subset of a Span is also a subset of that Span

Last updated Nov 1, 2022

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

This follows because every Vector Subspace of $V$ containing $R$ also contains $S$. Therefore we have $${W \subset V : S \subset W, W \text{ is a subspace of V}} \supset {W \subset V : R \subset W, W \text{ is a subspace of V}}.$$ Then $$\begin{align*} \text{span} S &= \bigcap\limits{W \subset V : S \subset W, W \text{ is a subspace of V}}\\ &\subset \bigcap\limits{W \subset V : R \subset W, W \text{ is a subspace of V}} & \text{ (fewer sets in the intersection)}\\ &=\text{span} R \end{align*}$$ establishing the result. $\blacksquare$

• Subspace Span