Zulip Chat Archive

Stream: general

Topic: zero vector space


view this post on Zulip Yury G. Kudryashov (Jan 03 2020 at 23:14):

Hi, is there any canonical way to say "a vector space is either {0}, or has a non-zero element"? I mean, are there any lemmas in linear_algebra that take some specific way of saying "E is not {0}" as an argument?

view this post on Zulip Alexander Bentkamp (Jan 05 2020 at 15:20):

I don't know what the canonical way to say it is, but maybe you could use these to lemmas that I proved recently, but haven't merged into mathlib yet:

lemma exists_mem_ne_zero_of_dim_pos' {α : Type v} {β : Type w}
  [discrete_field α] [add_comm_group β] [vector_space α β]
  (h_dim : 0 < dim α β) :  x : β, x  0 :=
begin
  obtain b, _, _⟩ : ( b : β, b  ( : submodule α β)  b  0),
  { apply exists_mem_ne_zero_of_dim_pos,
    rw dim_top,
    apply h_dim },
  use b
end

lemma dim_pos_of_mem_ne_zero {α : Type v} {β : Type w}
  [discrete_field α] [add_comm_group β] [vector_space α β]
  (x : β) (h : x  0) : 0 < dim α β :=
begin
  classical,
  by_contra hc,
  rw [not_lt, cardinal.le_zero, dim_top] at hc,
  have x_mem_bot : x  ,
  { rw  submodule.bot_of_dim_zero  hc,
    apply mem_top },
  exact h ((mem_bot α).1 x_mem_bot)
end

Last updated: May 11 2021 at 00:31 UTC