Definition talk:Dimension of Vector Space/Finite

Clapham is imprecise on the matter, which is why the initial definition suggested $n \in \N_{>0}$, but the question needs to be settled as to what it actually means to be a zero-dimensional vector space. --prime mover (talk) 17:18, 1 January 2023 (UTC)

It's true we do not have a results about the zero-dimensional vector space, as least I could not found any. What we do have is Sufficient Conditions for Basis of Finite Dimensional Vector Space, which says that:

"Let $n \ge 0$ be a natural number.

Let $E$ be an $n$-dimensional vector space over $K$."

implying that $0$ is a possible dimension. We also have Empty Set is Linearly Independent, which we could use to prove that the empty set $\O$ is a basis for a vector space.

Should I upload a theorem like Existence of Zero-Dimensional Vector Space?

BTW, happy new year, and thanks for your guidance, as always. --Anghel (talk) 18:38, 1 January 2023 (UTC)

thank you for your vote of confidence, and a happy new year to you

Existence of Zero-Dimensional Vector Space would be a suggestion, along with an entry in Also see for a start.

Once clarification of what this concept means, perhaps add a subpage referencing the degenerate case (if it actually is technically a degenerate case). --prime mover (talk) 18:47, 1 January 2023 (UTC)

Nevermind, I discovered we have such a theorem. It is called Trivial Vector Space iff Zero Dimension. I have changed the category of the theorem to Category:Dimension of Vector Space, so it should be easier to find now. --Anghel (talk) 21:53, 1 January 2023 (UTC)

Yes of course, that works. That ties off that loose thread. --prime mover (talk) 22:25, 1 January 2023 (UTC)