Covariance and Contravariance of Vectors - Definition

Definition

The general formulation of covariance and contravariance refers to how the components of a coordinate vector transform under a change of basis (passive transformation). Thus let V be a vector space of dimension n over the field of scalars S, and let each of f = (X₁,...,X_n) and f' = (Y₁,...,Y_n) be a basis of V. Also, let the change of basis from f to f′ be given by

(1)

for some invertible n×n matrix A with entries . Here, each vector Y_j of the f' basis is a linear combination of the vectors X_i of the f basis, so that