2
$\begingroup$

I was taught that every vector space has at least two subspaces: itself and its zero subspace. Does this still hold true for the zero vector space? You would think it would only have one subspace: itself, because it is also the zero subspace.

$\endgroup$
6
  • $\begingroup$ You are correct. If it were literally true that every vector space has at least two subspaces, then every vector space would have at least one proper subspace and there would be an infinite descending chain of subspaces... which isn't possible for a finite-dimensional vector space :). $\endgroup$
    – Erick Wong
    Mar 21, 2017 at 20:08
  • $\begingroup$ You can also ask, how many bases a vector space has, in particular the zero space. $\endgroup$ Mar 21, 2017 at 20:10
  • $\begingroup$ @Dietrich Burde Wouldn't the zero vector space have 0 bases, as the only vector in it is 0, and containing 0 would mean it is linearly dependent? $\endgroup$
    – WaveX
    Mar 21, 2017 at 20:20
  • $\begingroup$ @WaveX This is impossible, you know that every vector space has a basis. $\endgroup$ Mar 21, 2017 at 20:33
  • $\begingroup$ @DietrichBurde My apologies. This topic is still very new to me. $\endgroup$
    – WaveX
    Mar 21, 2017 at 20:34

1 Answer 1

4
$\begingroup$

The two subspaces in question here are the same, so the zero space really has one subspace - itself.

$\endgroup$
2
  • 1
    $\begingroup$ So with the exception of the zero vector space, all the rest have at least two, correct? $\endgroup$
    – WaveX
    Mar 21, 2017 at 20:06
  • 1
    $\begingroup$ This is correct. Of course, these subspaces are trivial. $\endgroup$ Mar 21, 2017 at 20:07

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .