In ℝⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.

1961 [Prentice-Hall], Richard A. Silverman (translator), Georgi E. Shilov, An Introduction to the Theory of Linear Spaces, 1974, Dover, page 66,
A more general linear form in the same space is the expression
f(x)=∑ₖ₊₁ⁿc_kζₖ
with arbitrary fixed coefficients c_1,c_2…,c_n.

In the theory of distributions, functions are replaced by linear forms on an auxiliary vector space, whose members are called test functions.

In order to better understand the effect of reducing a K-vector subspace V⊆P_d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.

Linear form

Meaning

Noun

Synonyms