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For a set to be a vector space, it needs to be:

  1. Closed under addition.
  2. Closed under scalar multiplication.

If I'm correct, the zero vector satisfies these two conditions:

  1. $0+0=0$
  2. $c\cdot 0=0$

Hence, my question narrows down to:

Is the zero vector itself considered a vector space? Or is a non-empty vector space considered to be the zero vector plus some other vectors?

Thanks in advance.

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    $\begingroup$ Yes it is a vector space. $\endgroup$ Commented Nov 9, 2014 at 19:06
  • $\begingroup$ It's a vector space over any field, even. (I think that means it's the best vector space) $\endgroup$ Commented Nov 9, 2014 at 19:07

2 Answers 2

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Yes. It is the zero space, which is zero-dimensional and which consist of one (very lonely) element.

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As other people have said it is a vector space but I think it might be good to provide an example of why we would want it to be one.

The first isomorphism theorem for vector spaces states that the kernel and image of a linear map are subspaces of the source and target respectively. If we take the identity map $i:V \rightarrow V$ then we know that the only element in $\ker (i)$ is $0$ (because the map is one to one). So if we want to easily state this theorem it helps to have $0$ count as a vector space. Similarly the map $f: V \rightarrow V$ with $f(v) = 0$ for all $v$ is linear and has $0$ as its image so once again we would want $0$ to be a vector space to state the theorem simply.

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