covariant and contravariant

Consider a vector space \(V\) over field \(F\).

Vectors are contravariant. A vector can be represented using coordinates with respect to a basis: \[ v = c_1 \mathbf{e_1} + \cdots + c_n\mathbf{e_n} \] When we apply a change of basis \(R\) to the basis vectors, the coordinates change by \(R^{-1}\). Why? Imagine that we do a change of basis that makes all basis vectors two times longer. Then, the coordinates of the same vector should now be two times smaller to compensate. Think of going from meters to centimeters. A coordinate of 3 meters is now a coordinate of 0.3 centimeters.

Covectors, which correspond with linear maps \(w: V \rightarrow F\) are covariant with the change of basis. This happens to compensate for the contravariant change to the vectors. Consider a covector \(w\). So. We apply a change of basis, all the vectors get shorter, but we still want our map \(w\) to map all the same vectors to the same scalars in the new coordinate description. So we need to apply \(R\) to \(w\) so we get that for a given vector \(v\), in the new basis we get \((w R)(R^{-1} v) = w\cdot v\), which will be the same, no matter what change of basis \(R\) we apply.