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:

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)