Can anybody explain why the vector bundle corresponding to a locally free sheaf F is global spec of sym of the dual of F and not just F? How does a section get identified with a polynomial in the dual?
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Let $L$ be a locally free ${\cal O}_X$module of finite rank. Define $V=Spec(Sym(L^\vee))$. Then $$Mor_X(X, V)={\cal O}_XAlg(Sym(L^\vee), {\cal O}_X)=Hom(L^\vee, {\cal O}_X)=L(X).$$ The universal mapping property of the (global) Spec is in EGA II.1. 

