This book is a short primer in engineering mathematics with a view on applications in nonlinear
control theory. In particular it introduces some elementary concepts of commutative algebra
and algebraic geometry which offer a set of tools quite different from the traditional
approaches to the subject matter. This text begins with the study of elementary set and map
theory. Chapters 2 and 3 on group theory and rings respectively are included because of their
important relation to linear algebra the group of invertible linear maps (or matrices) and the
ring of linear maps of a vector space. Homomorphisms and Ideals are dealt with as well at this
stage. Chapter 4 is devoted to the theory of matrices and systems of linear equations. Chapter
5 gives some information on permutations determinants and the inverse of a matrix. Chapter 6
tackles vector spaces over a field Chapter 7 treats linear maps resp. linear transformations
and in addition the application in linear control theory of some abstract theorems such as the
concept of a kernel the image and dimension of vector spaces are illustrated. Chapter 8
considers the diagonalization of a matrix and their canonical forms. Chapter 9 provides a brief
introduction to elementary methods for solving differential equations and finally in Chapter
10 nonlinear control theory is introduced from the point of view of differential algebra.
