This self-contained, clearly written textbook on linear algebra is
easily accessible for students. It begins with the simple linear
equation and generalizes several notions from this equation for the
system of linear equations and introduces the main ideas using matrices.
It then offers a detailed chapter on determinants and introduces the
main ideas with detailed proofs. The third chapter introduces the
Euclidean spaces using very simple geometric ideas and discusses various
major inequalities and identities. These ideas offer a solid basis for
understanding general Hilbert spaces in functional analysis. The
following two chapters address general vector spaces, including some
rigorous proofs to all the main results, and linear transformation:
areas that are ignored or are poorly explained in many textbooks.
Chapter 6 introduces the idea of matrices using linear transformation,
which is easier to understand than the usual theory of matrices
approach. The final two chapters are more advanced, introducing the
necessary concepts of eigenvalues and eigenvectors, as well as the
theory of symmetric and orthogonal matrices. Each idea presented is
followed by examples.
The book includes a set of exercises at the end of each chapter, which
have been carefully chosen to illustrate the main ideas. Some of them
were taken (with some modifications) from recently published papers, and
appear in a textbook for the first time. Detailed solutions are provided
for every exercise, and these refer to the main theorems in the text
when necessary, so students can see the tools used in the solution.
