I Vector Spaces.- §1. Definitions.- §2. Bases.- §4. Sums and Direct Sums.- II Matrices.- § 1. The Space of Matrices.- §2. Linear Equations.- §3. Multiplication of Matrices.- III Linear Mappings.- § 1. Mappings.- §2. Linear Mappings.- §3. The Kernel and Image of a Linear Map.- §4. Composition and Inverse of Linear Mappings.- §5. Geometric Applications.- IV Linear Maps and Matrices.- § 1. The Linear Map Associated with a Matrix.- §2. The Matrix Associated with a Linear Map.- §3. Bases, Matrices and Linear Map.- V Scalar Products and Orthogonality.- § 1. Scalar Products.- §2. Orthogonal bases, Positive Definite Case.- §3. Application to Linear Equations; the Rank.- §4. Bilinear Map and Matrices.- §5. General Orthogonal Bases.- §6. The Dual Space and Scalar Products.- §7. Quadratic Forms.- §8. Sylvester's Theorem.- VI Determinants.- §2. Existence of Determinants.- §3. Additional Properties of Determinants.- §4. Cramer's rule.- §5. Triangulation of a Matrix by Column Operations.- §6. Permutations.- §7. Expansion Formula and Uniqueness of Determinants.- §8. Inverse of a Matrix.- §9. The Rank of Matrix and Subdeterminants.- VII Symmetric, Hermitian and Unitary Operators.- §1. Symmetric Operators.- §2. Hermitian Operators.- §3. Unitary Operators.- VIII Eigenvectors and Eigenvalues.- §1. Eigenvectors and Eigenvalues.- §2. The Characteristic Polynomial.- §3. Eigenvalues and Eigenvectors of Symmetric Matrices.- §4. Diagonalization of a Symmetric Linear Map.- §5. The Hermitian Case.- IX Polynomials and Matrices.- §2. Polynomials of Matrices and Linear Maps.- X Triangulation of Matrices and Linear Maps.- §1. Existence of Triangulation.- §3. Diagonalization of Unitary Maps.- XI Polynomials and Primary Decomposition.- §1. The EuclideanAlgorithm.- §2. Greatest Common Divisor.- §3. Unique Factorization.- §4. Application to the Decomposition of a Vector Space.- §5. Schur's Lemma.- §6. The Jordan Normal Form.- XII Convex Sets.- §4. The Krein-Milman Theorem.- APPENDIX Complex Numbers.

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.