Editing Hodge star operator (section)

== Geometric explanation ==
The Hodge star is motivated by the correspondence between a subspace {{math|''W''}} of {{math|''V''}} and its orthogonal subspace (with respect to the scalar product), where each space is endowed with an [[Orientation (vector space)|orientation]] and a numerical scaling factor. Specifically, a non-zero decomposable {{math|''k''}}-vector <math>w_1\wedge\cdots\wedge w_k\in \textstyle\bigwedge^{\!k} V</math> corresponds by the [[Plücker embedding]] to the subspace <math>W</math> with oriented basis <math>w_1,\ldots,w_k</math>, endowed with a scaling factor equal to the {{math|''k''}}-dimensional volume of the parallelepiped spanned by this basis (equal to the [[Gramian matrix|Gramian]], the determinant of the matrix of scalar products <math>\langle w_i, w_j \rangle</math>). The Hodge star acting on a decomposable vector can be written as a decomposable ({{math|''n'' − ''k''}})-vector:
: <math>{\star}(w_1\wedge\cdots\wedge w_k) \,=\, u_1\wedge\cdots\wedge u_{n-k},</math>
where <math>u_1,\ldots,u_{n-k}</math> form an oriented basis of the [[Orthogonality|orthogonal space]] <math>U = W^\perp\!</math>. Furthermore, the ({{math|''n'' − ''k''}})-volume of the <math>u_i</math>-parallelepiped must equal the {{math|''k''}}-volume of the <math>w_i</math>-parallelepiped, and <math>w_1,\ldots,w_k,u_1,\ldots,u_{n-k}</math> must form an oriented basis of <math>V</math>.

A general {{math|''k''}}-vector is a linear combination of decomposable {{math|''k''}}-vectors, and the definition of Hodge star is extended to general {{math|''k''}}-vectors by defining it as being linear.