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Assume $S\subseteq U$ where $S = \emptyset$ and $U$ a vector space. Why is $S$ a subspace automatically if it doesn't contain the zero vector. I understand that it satisfies the other two properties to be a subspace, but the zero vector just doesn't fit.

EDIT:

My inquiry came from the following problem whose proof is stated as below:

If V is a vector space and S a subset of V then the span of S is a subspace of V.

Beginning of proof: if S is the empty set then this is clear.....

My thoughts: however if $S$ is empty then the span of S is also empty..... so the empty set is a subspace what????

daniel
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1 Answers1

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The proof in your edit is not claiming the empty set is a subspace. It is claiming the span of the empty set is a subspace (namely $\{0\}$). So, the original claim still holds even when $S$ is the empty set.

angryavian
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  • How can the span of an empty set contain the zero vector? – daniel Aug 12 '18 at 01:57
  • I don't see how the span of $\emptyset$ can be $\left{\vec 0\right}$. What's the definition of 'span' in that source? – Alejandro Nasif Salum Aug 12 '18 at 02:14
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    @daniel This is ultimately a convention about how sums work; see this wiki page. It can be weird at first, but it turns out to be useful (and it's clearly the only reasonable way to define the empty sum; similarly, the empty product is $1$, the union of no sets is the emptyset, and the intersection of no sets is the whole universe). – Noah Schweber Aug 12 '18 at 02:17
  • @AlejandroNasifSalum See my comment. – Noah Schweber Aug 12 '18 at 02:17
  • Ok. I knew the convention in other contexts, but I had never thought about it for vector spaces. Thanks – Alejandro Nasif Salum Aug 12 '18 at 02:29
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    @daniel Another definition for $\text{span}(A)$ is "the intersection of all subspaces containing $A$," from which $\text{span}(\varnothing) = {0}$ follows immediately. – angryavian Aug 12 '18 at 03:21