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Inhaltsangabe
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.
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.
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