Dual vector

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 $V\rightarrow V^{*}:v\mapsto v^{*}$:

$v^{*}(w):=v\cdot w\,$

and back:$V^{*}\rightarrow V:f\mapsto f^{*}$:

$f^{*}$ is the unique element of V for which for all $w\in V$

$f^{*}\cdot w=f(w).\,$

This isomorpism is involutive, that is $(v^{*})^{*}=v$

The above defined vector $v^{*}\in V^{*}$ is said to be the dual vector of $v\in V$.

See also

• Dual basis
• One-form