Linear Functional - Dual Vectors and Bilinear Forms

Dual Vectors and Bilinear Forms

See also: Hodge dual

Every non-degenerate bilinear form on a finite-dimensional vector space V gives rise to an isomorphism from V to V*. Specifically, denoting the bilinear form on V by <, > (for instance in Euclidean space <v,w> = vw is the dot product of v and w), then there is a natural isomorphism given by

The inverse isomorphism is given by where ƒ* is the unique element of V for which for all wV

The above defined vector v* ∈ V* is said to be the dual vector of vV.

In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem. There is a mapping VV* into the continuous dual space V*. However, this mapping is antilinear rather than linear.

Read more about this topic:  Linear Functional

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