Dual vector
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In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. If V is finite-dimensional, then V is isomorphic to V*.
Every non-degenerate bilinear product on a finite-dimensional space gives rise to an isomorphism from V to V*.
Specifically, in Euclidean spaces the dot product provides a natural isomorphism :
and back::
is the unique element of V for which for all
This isomorpism is involutive, that is
The above defined vector is said to be the dual vector of .