This book provides a comprehensive advanced multi-linear algebra course
based on the concept of Hasse-Schmidt derivations on a Grassmann algebra
(an analogue of the Taylor expansion for real-valued functions), and
shows how this notion provides a natural framework for many ostensibly
unrelated subjects: traces of an endomorphism and the Cayley-Hamilton
theorem, generic linear ODEs and their Wronskians, the exponential of a
matrix with indeterminate entries (Putzer's method revisited), universal
decomposition of a polynomial in the product of two monic polynomials of
fixed smaller degree, Schubert calculus for Grassmannian varieties, and
vertex operators obtained with the help of Schubert calculus tools
(Giambelli's formula). Significant emphasis is placed on the
characterization of decomposable tensors of an exterior power of a free
abelian group of possibly infinite rank, which then leads to the
celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP)
hierarchy describing the Plücker embedding of an infinite-dimensional
Grassmannian. By gathering ostensibly disparate issues together under a
unified perspective, the book reveals how even the most advanced topics
can be discovered at the elementary level.