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In class today, we talked about vector subspaces.

A subspace $v$ of a vector space $V$ (for this instance, $V$ is $\mathbb R^3$) is a space which all values of $v$ are within $V$, and some vector within $V$ can be added to $v$ to get another specific vector which is contained within $V$, thus every vector in $V$ can be made by summing some multiples of vectors within $v$

Say that $v$ is $\{[1\;0\;0],[0\;1\;0],[0\;0\;1]\}$, the column vectors of $Id_3$.

My professor said that the empty set $\emptyset = \{\}$ is within any set that spans $\mathbb R^3$.

This means that there is a sum of multiplies of the vectors within v that is equal to $\emptyset$, correct? Why or why not?

Asaf Karagila
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tuskiomi
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  • The empty set $\emptyset$ is a subset of any set. For a vector space, we also have the set containing the zero vector is contained in the vector space, that is ${0} \subset V$. However, these are two completely different things. – copper.hat Oct 17 '17 at 18:33
  • If your professor really did use $v$ for the name of a subspace of a vector space $V$, then it's no wonder you're confused. I would be too. – Dustan Levenstein Oct 17 '17 at 18:33
  • @DustanLevenstein i couldn't figure out how to serif the V. – tuskiomi Oct 17 '17 at 18:34
  • Your main paragraph (the second one) in a single sentence describes three different things (and the middle one doesn't even make much sense). Could you please explain more clearly what exactly you're talking about and what exactly you're asking? – zipirovich Oct 17 '17 at 18:37
  • @zipirovich i wish that i could. Maybe if you tell me what 3 things I'm referring to, i could. – tuskiomi Oct 17 '17 at 18:39
  • v is not a subspace it's just a set that spans V (a basis for V) – shvd1991 Oct 17 '17 at 18:42
  • @DustanLevenstein I3 is the 3x3 identity matrix. – tuskiomi Oct 17 '17 at 18:43
  • and also empty set is not in every set that spans V. it's so not correct – shvd1991 Oct 17 '17 at 18:43
  • @Parto why not? – tuskiomi Oct 17 '17 at 18:47
  • @tuskiomi :"My professor said that the empty set is within any set that spans R3". It means that empty set is within every basis of R3. it's not correct so maybe edit your post. – shvd1991 Oct 17 '17 at 18:50
  • @tuskiomithe correct sentence would be empty set is in the set of all subsets of a set. – shvd1991 Oct 17 '17 at 18:53
  • @parto but that's one of the properties of the empty set, that it's a subset of every set, thus there is a null element in every set. – tuskiomi Oct 17 '17 at 18:54
  • @tuskiomi Note that a subset of a set is not the same thing as an element of a set. For example, $\emptyset$ is a subset of the set ${1}$, but it is not an element of ${1}$. – Dustan Levenstein Oct 17 '17 at 18:58
  • @DustanLevenstein as far as I see,${}$ doesn't occur in ${1}$, either. – tuskiomi Oct 17 '17 at 18:59
  • Can you find an element of ${}$ which is not an element of ${1}$? If you can't, then it's a subset. – Dustan Levenstein Oct 17 '17 at 19:00
  • @DustanLevenstein the element "" is not a part of the list. – tuskiomi Oct 17 '17 at 19:01
  • @tuskiomi The intended meaning of ${}$ is a set which has no elements. It seems you're reading that it does contain an element, namely the empty string. That is not correct. – Dustan Levenstein Oct 17 '17 at 19:03

2 Answers2

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The empty set is a subset, not a member of every such set. Only if the empty set were a member of the span of a set of vectors would it be a sum of scalar multiples of those vectors. The empty set is not a vector, so it's not a sum of scalar multiples of any vectors.

Michael Hardy
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I think there may have been some missing information in what your professor told you.

From a set theory standpoint, if $U$ is any subspace of a vector space $V$ then $\emptyset \subset U \subseteq V.$ Now, you might wonder what role $\emptyset$ plays in the language of linear algebra.

The trivial subspace $\{0\}$ is a subspace of every vector space, which you can verify axiomatically that it satisfies all of the conditions of a subspace. You might also recognize that $\emptyset$ is a basis for $\{0\}/$ and so $\{0\}$ is a subspace of any vector space because its basis, namely $\emptyset,$ is subset of every basis, for any vector space.

Chickenmancer
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    It is not true that $\emptyset$ is a basis for ${0}$. – copper.hat Oct 17 '17 at 18:59
  • @copper.hat seriously? Even if you have some gripes with empty sums or something, you have to acknowledge that plenty of mathematicians agree that the empty set is a basis for ${0}$. – Dustan Levenstein Oct 17 '17 at 19:14
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    @copper.hat, Would you care to explain why it is not a basis? – Chickenmancer Oct 17 '17 at 19:23
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    There are many resources which agree with my assertion being correct. I'm happy to change it if you have a resource which disagrees with my assertion. Example: https://math.stackexchange.com/questions/1909645/empty-set-is-the-only-basis-of-the-zero-vector-space-0 – Chickenmancer Oct 17 '17 at 19:26
  • How can I choose vectors in $\emptyset$ such that $0$ is a linear combination of those non existent vectors? – copper.hat Oct 17 '17 at 20:12
  • @DustanLevenstein: I realise that it is a convention that many use, but in context I think it very confusing for the OP. – copper.hat Oct 17 '17 at 20:18