linear form

linear form

English

Noun

linear form (plural linear forms)

1. (linear algebra) A linear functional.

   In ${\\displaystyle \\mathbb {R} ^{n}}$, 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

     ${\\displaystyle f(x)=\\sum _{k+1}^{n}{c_{k}\\zeta _{k}}}$

     with arbitrary fixed coefficients ${\\displaystyle c_{1},c_{2}\\dots ,c_{n}}$.

   • 1998, F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd Edition, Cambridge University Press, page 2,

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

   • 2005, Martin Kreuzer, Lorenzo Robbiano, Computational Commutative Algebra 2, Springer, page 264,

     In order to better understand the effect of reducing a ${\\displaystyle K}$-vector subspace ${\\displaystyle V\\subseteq P_{d}}$ modulo a generic linear form, we generalize Proposition 5.5.18 as follows.

Usage notes

• The terms linear form and linear functional are semantically interchangeable. However, the former appears to emphasise that the entity is an algebraic structure, while the latter emphasises that it is a mapping. Thus, linear form is arguably more appropriate to linear algebra, and linear functional to functional analysis.

Synonyms

• (linear functional): covector, linear functional, one-form

Derived terms

• linear form in logarithms

Related terms

• bilinear form
• multilinear form
• differential form
• polynomial form

See also

• dual space

Further reading

• Linear Functional on Wolfram MathWorld