Beyond Monge's Theorem: A Generalization of the Pythagorean Theorem

Abstract

It is a well-known fact from linear algebra that the of an n-dimensional parallelepiped in lR spanned by vectors A1, . . ., An is given by the (absolute value of the) determinant of the matrix with the vectors Al,... An as rows or columns. Although perhaps not all students realize this, it is this property that is behind those pesky integral transformations in (advanced) calculus that one has to go through to determine the of almost any object in 3-space that is more complicated than a sphere or a cylinder. But what if, rather than a parallelepiped that has the same dimensions as the space it lives in, we have a parallelepiped that is spanned by fewer vectors? In the linear space spanned by the vectors that define such a parallelepiped, this parallelepiped will have a as well, but there does not seem an easy way to determine such lower-dimensional volume. In other words, if instead of our n-dimensional parallelepiped living in R , we have a p-dimensional parallelepiped in RO, how can the (p-dimensional) of that parallelepiped be determined? Let us consider some examples to make it clear what we are looking for. If we have just one vector in R n (n > 1), our parallelepiped will be a line segment, the (1-dimensional) volume of which is usually referred to as length. In this case, for any point A = (a,,.. . an) in R , the length of the corresponding vector A is defined by the relation