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.
