Let Λ be an algebra over R The exterior algebra Λ_{n}^{∗} is the algebra with product ∧ generated by v_{1}, ..., v_{n} with unit such that v_{i} ∧ v_{j} = −v_{j} ∧ v_{i} Let V be a finite-dimensional vector space over R. The dual of V = V^{∗} = {f : V → R | f linear } Note V^{∗} is a vector space. The elements of V^{∗} are called covectors. If e_{1}, ..., e_{n} basis for V, then w_{1}, ..., w_{n} basis for V^{∗} where w_{i} : V → R where

w_{i}(e_{j}) = δ_{ij} = \{ \begin{matrix}1 & i = j \\ 0 & i ≠ j \end{matrix}

Let F_{∗} : V → W be a linear map between vector spaces The dual map map is F^{∗} : W^{∗} → V^{∗}, F (g) = g ∘ F. F_{∗} is injective implies F^{∗} injective F_{∗} is surjective implies F^{∗} surjective (G_{∗} ∘ F_{∗})^{∗} = F^{∗} ∘ G^{∗}. d : V → (V^{∗})^{∗}, d(v) = h where h : V^{∗} → R, h(f) = f(v). Thus (V^{∗})^{∗} is naturally isomorphic to V. Defn: The dual of T_{p}M = T_{p}^{∗}M is the cotangent space to M at p. If \frac{∂}{∂x_{1}} , ..., \frac{∂}{∂x_{m}} is a basis for T_{p}M, then the dual basis will be denoted dx_{1}, ..., dx_{m}.