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I have a somewhat trivial question regarding clarification of something my textbook said. We are given the following definition of a subspace in a vector space. I believe my textbook is only talking about vector spaces in terms of scalar multiplication = all real numbers.

"A nonempty subset W of a vector space V is a subspace of V when W is a vector space under the operations of addition and scalar multiplication defined in V. "

Therefore, an empty subset W of a vector space V will not be a subspace of V under the addition and scalar multiplication defined in V.

My questions:

1) What is the difference between a 'subset' and a 'subspace'?

2) Is an empty subset just considered: empty_subset = {} ?

If so, empty_subset will NOT be equivalent to the zero vector subset: zero_vector_subset = {0} ?

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  • $\begingroup$ A subspace is a subset which is also a vector space. Since every vector space includes a zero element, the empty set cannot be a subspace. $\endgroup$ – Bungo Oct 23 '18 at 17:25
  • $\begingroup$ Because the empty set does not contain 0? That makes sense, thanks! $\endgroup$ – Evan Kim Oct 23 '18 at 17:27
  • $\begingroup$ Correct, the empty set does not contain 0, so it is not a subspace. $\endgroup$ – Bungo Oct 23 '18 at 17:28
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  1. A subspace, in this context, is a vector subspace. A subspace is a subset, bot most subsets are nt subspaces.
  2. There is only one empty set, denoted by $\emptyset$. In can also denote it by $\{\}$, but that's unusual

And, yes, $\{0\}$ (which is a vector subspace) is not the same thing as $\emptyset$.

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  • $\begingroup$ Where in mathematics (or what stage of linear algebra) does the empty set = ∅ come into use? $\endgroup$ – Evan Kim Oct 23 '18 at 17:28
  • $\begingroup$ Whenever you deal with sets, you hve to deal with the possibility that two sets don't intersect. Therefore, you will need the empty set to express their intersection. $\endgroup$ – José Carlos Santos Oct 23 '18 at 17:32
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1) Subset has just set-theoretical meaning. While in this context, a subspace is again a vector space carrying an algebraic structure.

2) Yes. $\emptyset$ is different than $\{ 0 \}$ because it has an element.

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A subspace $V\subset W$ of a vector space $W$ is a subset that is a vector space itself with the operation being those of $W$ restricted to $V$. Recall that a vector space $W$ is an Abelian group together with scalar multiplication. A subspace $V$ now is an Abelian subgroup $V\subset W$ of this group together with the restriction of the scalar multiplication on $W$ to $V$.

Every Abelian group needs to have a zero-element. Thus, the empty set is not an Abelian group and therefore can't be given the structure of a vector space.

(Ok, I see that other people were much faster typing their answers. Anyway I hope this still helps. I find it important to emphasize on the fact that the subspace structure is induced by the larger vector space and to also give an explanation why the empty set can never be a vector space.)

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  • $\begingroup$ That is an interesting property of an empty set, thanks for sharing $\endgroup$ – Evan Kim Oct 23 '18 at 18:02

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