In the following, V is a vector space over an arbitrary field F, dimFV = n. Let {e1, . . . , en} be a basis for V, and {f1, . . . , fn} be the dual basis for and , then the operators *(u) and i(g) (exterior and inner multiplication by u and g respectively) set up an equivalence between the ideal = range of ∊(u) and the sub-algebra = range of i(g) considered as vector spaces. That is, ∊(u)i(g) is the identity on , i(g) ∊(u) is the identity on .