content_image/318092.jpg content_image/318091.jpg content_image/318090.jpg
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 

<img src='content_image/318092.jpg'>

<img src='content_image/318091.jpg'>

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 

<img src='content_image/318090.jpg'>

