and note that any 1-covector can be written in this basis. Definition 1.3.2 (Exterior product, simple covectors) . Given a$_{1}$, . . . , a$_{m}$ ∈ ∧ $^{1}$( R $^{n}$) , we denote the exterior or wedge product of these 1-covectors to be the m -covector which is defined by where the vector a$_{i}$ is the representation of the 1-covector a$_{i}$ in the dual basis [d x$_{1}$, . . . , d x$_{n}$ ] . Any element of ∧ $^{m}$( R $^{n}$) that can be written as a wedge product of 1-covectors a$_{i}$ is called simple and every m -covector can be expressed as the sum of simple m -covectors. The wedge product extends to higher degree covectors by means of this decomposition and a distributive law. The wedge product is m -linear and is negated when any two covectors are transposed because it relies on the determinant. For the same reason, if a particular 1-covector appears more than once in the wedge product, the result is 0. Working an example, we have 5 (4 d x$_{1}$ ∧ d x$_{3}$ + 3 d x$_{4}$ ∧ d x$_{3}$ ) ∧ (2 d x$_{1}$ ∧ d x$_{2}$ − d x$_{1}$ ∧ d x$_{3}$ ) = 8 d x$_{1}$ ∧ d x$_{3}$ ∧ d x$_{1}$ ∧ d x$_{2}$ − 4 d x$_{1}$ ∧ d x$_{3}$ ∧ d x$_{1}$ ∧ d x$_{3}$ + 6 d x$_{4}$ ∧ d x$_{3}$ ∧ d x$_{1}$ ∧ d x$_{2}$ − 3 d x$_{4}$ ∧ d x$_{3}$ ∧ d x$_{1}$ ∧ d x$_{3}$ = 6 d x$_{4}$ ∧ d x$_{3}$ ∧ d x$_{1}$ ∧ d x$_{2}$. ( a$_{1}$ ∧ a$_{2}$ ∧ · · · ∧ a$_{m}$ )( u$_{1}$ , . . . , u$_{m}$ ) = det$^{}$         a$_{1}$ . . . a$_{m}$$^{}$    T     u$_{1}$ . . . u$_{m}$$^{}$         a$_{1}$ ∧ a$_{2}$ ∧ · · · ∧ a$_{m}$