span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2)

Luisito89

Hi! I wonder if you guys help me prove this:
span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2)
I have seen span(S1 U S2) = span(S1) + span(S2), so I just want to prove this one, any suggestion?

MathematicalPhysicist

v=as, s in S1 and S2, thus v is in sp(S1) and sp(S2).
Vice versa isn't necessarily because if v in sp(S1) and sp(S2), then v=as1=bs2, and s1 isn't necessarily in S2 and s2 isn't necessarily in S1.

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Luisito89	span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2) Tweet Hi! I wonder if you guys help me prove this: span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2) I have seen span(S1 U S2) = span(S1) + span(S2), so I just want to prove this one, any suggestion?

MathematicalPhysicist	v=as, s in S1 and S2, thus v is in sp(S1) and sp(S2). Vice versa isn't necessarily because if v in sp(S1) and sp(S2), then v=as1=bs2, and s1 isn't necessarily in S2 and s2 isn't necessarily in S1.