Does Sp(V)=Sp(W) mean that V=W?

Intuitively I sense it does not, but I cannot find the right arguments to reason it…

Could anybody help out?

  • 2
    $\begingroup$ In $\mathbb R^{2}$, $Span (1,0) =span (2,0)$ but $(1,0) \neq (2,0)$ $\endgroup$ – Kavi Rama Murthy Dec 10 '18 at 8:09
  • $\begingroup$ Thank you! so as long as W is a linear combination of V, then their spans are identical, despite the fact that the groups themselves are different? $\endgroup$ – Pregunto Dec 10 '18 at 8:19
  • $\begingroup$ Groups? Which groups? $\endgroup$ – José Carlos Santos Dec 10 '18 at 8:24
  • $\begingroup$ @JoséCarlosSantos groups V and W $\endgroup$ – Pregunto Dec 10 '18 at 8:30
  • 1
    $\begingroup$ Sorry, I am translating from my own native tongue! I guess the right word in English is indeed "sets"... $\endgroup$ – Pregunto Dec 10 '18 at 8:32

No. Take a basis, $V=\{v_1,\dots, v_n\}$, and then take a nontrivial scalar multiple of each basis element, call the resulting set $W=\{\alpha v_1,\dots, \alpha v_n\}$. Then $V\cap W=\emptyset$, but $\operatorname{span}V=\operatorname{span}W$.


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