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AuthorShakarchi, Rami. author
TitleSolutions Manual for Lang's Linear Algebra [electronic resource] / by Rami Shakarchi
ImprintNew York, NY : Springer New York : Imprint: Springer, 1996
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Descript XI, 200 p. online resource


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, ยง3 of Chapter II shows that a strictly upper triangular matrix is nilpotent and this result is then used in Exercise 7, ยง1 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


I Vector Spaces -- ยง1. Definitions -- ยง2. Bases -- ยง4. Sums and Direct Sums -- II Matrices -- ยง 1. The Space of Matrices -- ยง2. Linear Equations -- ยง3. Multiplication of Matrices -- III Linear Mappings -- ยง 1. Mappings -- ยง2. Linear Mappings -- ยง3. The Kernel and Image of a Linear Map -- ยง4. Composition and Inverse of Linear Mappings -- ยง5. Geometric Applications -- IV Linear Maps and Matrices -- ยง 1. The Linear Map Associated with a Matrix -- ยง2. The Matrix Associated with a Linear Map -- ยง3. Bases, Matrices and Linear Map -- V Scalar Products and Orthogonality -- ยง 1. Scalar Products -- ยง2. Orthogonal bases, Positive Definite Case -- ยง3. Application to Linear Equations; the Rank -- ยง4. Bilinear Map and Matrices -- ยง5. General Orthogonal Bases -- ยง6. The Dual Space and Scalar Products -- ยง7. Quadratic Forms -- ยง8. Sylvester's Theorem -- VI Determinants -- ยง2. Existence of Determinants -- ยง3. Additional Properties of Determinants -- ยง4. Cramer's rule -- ยง5. Triangulation of a Matrix by Column Operations -- ยง6. Permutations -- ยง7. Expansion Formula and Uniqueness of Determinants -- ยง8. Inverse of a Matrix -- ยง9. The Rank of Matrix and Subdeterminants -- VII Symmetric, Hermitian and Unitary Operators -- ยง1. Symmetric Operators -- ยง2. Hermitian Operators -- ยง3. Unitary Operators -- VIII Eigenvectors and Eigenvalues -- ยง1. Eigenvectors and Eigenvalues -- ยง2. The Characteristic Polynomial -- ยง3. Eigenvalues and Eigenvectors of Symmetric Matrices -- ยง4. Diagonalization of a Symmetric Linear Map -- ยง5. The Hermitian Case -- IX Polynomials and Matrices -- ยง2. Polynomials of Matrices and Linear Maps -- X Triangulation of Matrices and Linear Maps -- ยง1. Existence of Triangulation -- ยง3. Diagonalization of Unitary Maps -- XI Polynomials and Primary Decomposition -- ยง1. The Euclidean Algorithm -- ยง2. Greatest Common Divisor -- ยง3. Unique Factorization -- ยง4. Application to the Decomposition of a Vector Space -- ยง5. Schur's Lemma -- ยง6. The Jordan Normal Form -- XII Convex Sets -- ยง4. The Krein-Milman Theorem -- APPENDIX Complex Numbers

Mathematics Functions of real variables Mathematics Real Functions


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