Linear Algebra

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.