# Question about Span in Linear Algebra

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?

• In $\mathbb R^{2}$, $Span (1,0) =span (2,0)$ but $(1,0) \neq (2,0)$ – Kavi Rama Murthy Dec 10 '18 at 8:09
• 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? – Pregunto Dec 10 '18 at 8:19
• Groups? Which groups? – José Carlos Santos Dec 10 '18 at 8:24
• @JoséCarlosSantos groups V and W – Pregunto Dec 10 '18 at 8:30
• Sorry, I am translating from my own native tongue! I guess the right word in English is indeed "sets"... – 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 $$\operatorname{span}V=\operatorname{span}W$$.