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,...,Xn) and f' = (Y1,...,Yn) 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 Yj of the f' basis is a linear combination of the vectors Xi of the f basis, so that

Read more about this topic:  Covariance And Contravariance Of Vectors

Famous quotes containing the word definition:

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)