Triples u₁,u₂,u₃ and v₁,v₂,v₃ lie in ℝ⁴ and span U,V respectively. I:if dim(U∩V)=2 and dim U=3, the v triple is dependent. II:if U+V=ℝ⁴, at least one triple is independent. Which hold?
For I take U=span(e₁,e₂,e₃),V=span(e₁,e₂,e₄); intersection dimension is 2 but the v triple can be independent. For II take triples (e₁,e₂,e₁+e₂) and (e₃,e₄,e₃+e₄). Both are dependent yet their spans sum to ℝ⁴. Neither holds. **D**.
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