Dual Space - Algebraic Dual Space

Algebraic Dual Space

Given any vector space V over a field F, the dual space V* is defined as the set of all linear maps φ: VF (linear functionals). The dual space V* itself becomes a vector space over F when equipped with the following addition and scalar multiplication:

begin{align} & (varphi + psi)(x) = varphi(x) + psi(x) \ & (a varphi)(x) = a left(varphi(x)right) end{align}

for all φ, ψV*, xV, and aF. Elements of the algebraic dual space V* are sometimes called covectors or one-forms.

The pairing of a functional φ in the dual space V* and an element x of V is sometimes denoted by a bracket: φ(x) = or φ(x) = ⟨φ,x⟩. The pairing defines a nondegenerate bilinear mapping : V* × VF.

Read more about this topic:  Dual Space

Other articles related to "dual, algebraic dual space, space, dual space":

Dual Code
... In coding theory, the dual code of a linear code is the linear code defined by where is a scalar product ... In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form ... The dimension of C and its dual always add up to the length n A generator matrix for the dual code is a parity-check matrix for the original code and vice versa ...
Dual-use Technology
... Dual use goods are products and technologies normally used for civilian purposes but which may have military applications ... -European Commision, Trade Topics, Dual Use “ ” Dual-use is a term often used in politics and diplomacy to refer to technology which can be used for both peaceful and military aims ... More generally speaking, dual-use can also refer to any technology which can satisfy more than one goal at any given time ...
Algebraic Dual Space - Quotient Spaces and Annihilators
... The annihilator of a subset is itself a vector space ... a vector subspace, then after identifying W with its image in the second dual space under the double duality isomorphism V ≈ V** ... annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space ...
Dual - Media
... Dual (album), an album of traditional Scottish and Irish music recorded by Éamonn Doorley, Muireann Nic Amhlaoibh, Julie Fowlis and Ross Martin, released 2008 Dual (film) "Dual" (Heroes), an episode ...
A590 Road
... The road is a mixture of dual carriageway and single carriageway, with the section east of Low Newton, Cumbria to the M6 being mainly dual ... Further dual sections are south of Newby Bridge, south of Greenodd and south of Ulverston ...

Famous quotes containing the words space, algebraic and/or dual:

    In the tale proper—where there is no space for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.
    Edgar Allan Poe (1809–1849)

    I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
    Henry David Thoreau (1817–1862)

    Thee for my recitative,
    Thee in the driving storm even as now, the snow, the winter-day
    declining,
    Thee in thy panoply, thy measur’d dual throbbing and thy beat
    convulsive,
    Thy black cylindric body, golden brass and silvery steel,
    Walt Whitman (1819–1892)