Chapter 4 - Vector Spaces - 4.4 Spanning Sets - Problems - Page 284: 48

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Given $S$ and $S′$ are subsets of a vector space $V$ such that $S \subset S'$ Assume that $S=\emptyset\\ \rightarrow span (S)=\{0\}$ For $span(S')=\{0\} \\ \rightarrow span (S')=span (S)$ For $S' \ne 0 \rightarrow 0v=0 \rightarrow 0 \in span(S') \rightarrow span(S) \subset span(S')$ If $S\ne 0$, $v$ is a linear combination of vector in $S$ and since $S \subset S'$ we also have $v$ is a linear combination in $S'$ Hence, span $(S) \subset$span $(S')$