Editing Exterior algebra (section)

=== Linear algebra ===
In applications to [[linear algebra]], the exterior product provides an abstract algebraic manner for describing the [[determinant]] and the [[minor (matrix)|minors]] of a [[matrix (mathematics)|matrix]].  For instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation).  This suggests that the determinant can be ''defined'' in terms of the exterior product of the column vectors.  Likewise, the {{math|''k'' × ''k''}} minors of a matrix can be defined by looking at the exterior products of column vectors chosen {{math|''k''}} at a time.  These ideas can be extended not just to matrices but to [[linear transformation]]s as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope.  So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power.  The action of a transformation on the lesser exterior powers gives a [[basis of a vector space|basis]]-independent way to talk about the minors of the transformation.