Linear Algebra - Werner H. Greub

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Sommaire:
0. Prerequisites.- I. Vector spaces.- ? 1. Vector spaces.- ? 2. Linear mappings.- ? 3. Subspaces and factor spaces.- ? 4. Dimension.- ? 5. The topology of a real finite dimensional vector space.- II. Linear mappings.- ? 1. Basic properties.- ? 2. Operations with linear mappings.- ? 3. Linear isomorphisms.- ? 4. Direct sum of vector spaces.- ? 5. Dual vector spaces.- ? 6. Finite dimensional vector spaces.- III. Matrices.- ? 1. Matrices and systems of linear equations.- ? 2. Multiplication of matrices.- ? 3. Basis transformation.- ? 4. Elementary transformations.- 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.- V. Algebras.- ? 1. Basic properties.- ? 2. Ideals.- ? 3. Change of coefficient field of a vector space.- VI. Gradations and homology.- ? 1. G-graded vector spaces.- ? 2. G-graded algebras.- ? 3. Differential spaces and differential algebras.- 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.- 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. Differentiate families of linear automorphisms.- IX. Symmetric bilinear 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.- X. Quadrics.- ? 1. Affine spaces.- ? 2. Quadrics in the affine space.- ? 3. Affine equivalence of quadrics.- ? 4. Quadrics in the Euclidean space.- XI. Unitary spaces.- ? 1. Hermitian functions.- ? 2. Unitary spaces.- ? 3. Linear mappings of unitary spaces.- ? 4. Unitary mappings of the complex plane.- ? 5. Application to Lorentz-transformations.- XII. Polynomial algebra.- ? 1. Basic properties.- ? 2. Ideals and divisibility.- ? 3. Factor algebras.- ? 4. The structure of factor algebras.- 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 semisimple transformations.- ? 7. Applications to inner product spaces.