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Let $E$ be a vector space and $X$ a subset of $E$. Define the subspace $span(X)$ to be the intersection of all subspaces containing $X$. Let $C$ be the subspace of all linear combinations of elements of $X$. I want to show that $Span(X)=C$. One way is clear which is $C\subset Span(X)$ but I have difficulty showing the other way. For simplicity suppose $X=\{x_1,x_2\}$ and suppose $span(X)=F_1\cap F_2\cap F_3$ where $F_1$, $F_2$ and $F_3$ are the subspaces containing $X$. Take $x\in F_1\cap F_2\cap F_3$ why then should exist scalars $a_1$ and $a_2$ such that $x=a_1x_1+a_2x_2$? Thank you for your help!

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Show $C$ is a subspace which contains $X$. Then by definition of span$X$, span$X\subset C$.

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