The present volume contains all the exercises and their solutions of
Lang's' Linear Algebra. Solving problems being an essential part of the
learning process, my goal is to provide those learning and teaching
linear algebra with a large number of worked out exercises. Lang's
textbook covers all the topics in linear algebra that are usually taught
at the undergraduate level: vector spaces, matrices and linear maps
including eigenvectors and eigenvalues, determinants, diagonalization of
symmetric and hermitian maps, unitary maps and matrices, triangulation,
Jordan canonical form, and convex sets. Therefore this solutions manual
can be helpful to anyone learning or teaching linear algebra at the
college level. As the understanding of the first chapters is essential
to the comprehension of the later, more involved chapters, I encourage
the reader to work through all of the problems of Chapters I, II, III
and IV. Often earlier exercises are useful in solving later problems.
(For example, Exercise 35, §3 of Chapter II shows that a strictly upper
triangular matrix is nilpotent and this result is then used in Exercise
7, §1 of Chapter X.) To make the solutions concise, I have included only
the necessary arguments; the reader may have to fill in the details to
get complete proofs. Finally, I thank Serge Lang for giving me the
opportunity to work on this solutions manual, and I also thank my
brother Karim and Steve Miller for their helpful comments and their
support.
