This textbook gives a detailed and comprehensive presentation of linear algebra based on an axiomatic treatment of linear spaces. For this fourth edition some new material has been added to the text, for instance,the intrinsic treatment of the classical adjoint of a linear transformation in Chapter IV, as well as the discussion of quaternions and the classification of associative division algebras in Chapter VII. Chapters XII and XIII have been substantially rewritten for the sake of clarity, but the contents remain basically the same as before. Finally, a number of problems covering new topics- e.g. complex structures, Caylay numbers and symplectic spaces- have been added.

Chapter O. Prerequisites

Chapter Ⅰ. Vector spaces

 §1. Vector spaces

 §2. Linear mappings

 §3. Subspaces and factor spaces

 §4. Dhneasion.

 §5. The topology of a real finite dimensional vector space

Chapter Ⅱ. Linear mappings

 §1. Bask properties .

 §2. Operations with linear mappings

 §3. Linear isomorphisms

 §4. Direct sum of vector spaces

 §5. Dual vector spaces

 §6. Finite dimensional vector spaces

Chapter Ⅲ Matrices

 §1. Matrices and systems of linear equations

 §2. Multiplication of matrices

 §3. Basis transformation

 §4. Elementary transformations

Chapter Ⅳ. Determinants

 §1. Determinant functions

 §2. The determinant of a linear transformation

 §3. The determinant of a matrix

 §4. Dual determinant functions

 §5. The adjoint matrix

 §6. The characteristic polynomial

 §7. The trace

 §8. Oriented vector spaces

Chapter Ⅴ. Algebras

 §1. Basic properties

 §2. Ideals

 §3. Change of coeffient field of a vector space

Chapter Ⅵ. Gradations and homology

 §1. G-graded vector spaces

 §2. G-graded algebras

 §3. Differential spaces and differential alsebras

Chapter Ⅶ. Inner product spaces

 §1. The inner product

 §2. Orthonormal bases

 §3. Normed determinant functions

 §4. Duality in an inner product space

 §5. Normed vector spaces

 §6. The algebra of quaternions

Chapter Ⅷ. Linear mappings of inner product spaces

 §1. The adjoint mapping

 §2. Selfadjoint mappings

 §3. Orthogonal projections

 §4. Skew mappings

 §5. Isometric mappings

 §6. Rotations of Euclidean spaces of dimension 2, 3 and 4

 §7. Different~able families of linear automorphisms

Chapter Ⅸ. Symmetric bilinaar functions

 §1. Bilinear and quadratic functions

 §2. The decomposition of E

 §3. Pairs of symmetric bilinear functions

 §4. Pseudo-Euclidean spaces

 §5. Linear mappings of Pseudo-Euclidean spaces

Chapter Ⅹ. Quadries

 §1. Arlene spaces

 §2. Quadrics in the affme space

 §3. Affine equivalence of quadrics

 §4. Quadrics in the Euclidean space

Chapter Ⅺ. Unitary spaces

 §1. Hermitian functions

 §2. Unitary spaces

 §3. Linear mappings of unitary spaces

 §4. Unitary mappings of the complex Diane

 §5. Application to Lorentz-transformations

Chapter Ⅻ. Polynomial algebra

 §1. Basic properties

 §2. Ideals and divisibility

 §3. Factor algebras

 §4. The structure of factor algebras

Chapter ⅩⅢ. Theory of a linear transformation

 §1. Polynomials in a linear transformation

 §2. Generalized eigenspaces

 §3. Cyclic spaces

 §4. Irreducible spaces

 §5. Application of cyclic spaces

 §6. Nilpotent and sere~simple transformations

 §7. Applications to inner product spaces

Bibliography

Subject Index