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Given a finite set of affine spaces, all of which are subsets of a given vector space of finite dimension, is there any way to know if their union is the entire vector space, provided no additional assumptions are made? (For instance, they may or may not be disjoint).

This is equivalent to deciding if there is a vector in the space outside all spaces together, and that question was asked here (Finding a vector that isn't in a set of subspaces), but I'm not sure if, for instance, $\mathbb R^3$ is the union of the two proper affine subspaces spanned by the sets of vectors $\lbrace (1, 0, 0),\: (0, 1, 0) \rbrace$ and $\lbrace (0, 0, 1), \: (0, 1, 0) \rbrace$, which seems not to be consistent with the first answer.

I apologize in advance if this question is too basic or is phrased haphazardly!

Kipirpo
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  • It's a repeat. Over any field $F$, a nonzero finite dimensional vector space cannot be a union of fewer than $|F|$ proper subspaces. In particular, $\mathbb{R}^3$ cannot be the union of fewer than uncountably many proper subspaces. $\mathbb{R}^3$ is not the union of those two subspaces, as neither subspace contains the vector $(1,1,1)$. – Arturo Magidin Oct 30 '22 at 23:53
  • Oh, I see it now. Thanks! I'll leave the question as is for a couple of hours, then delete it so it does not clutter the site with redundant material – Kipirpo Oct 31 '22 at 17:33
  • It's been closed as a duplicate. It doesn't "clutter the site" any more. – Arturo Magidin Oct 31 '22 at 17:40

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