Linear Algebra

0. Prerequisites.- I. Vector spaces.- x 1. Vector spaces.- x 2. Linear mappings.- x 3. Subspaces and factor spaces.- x 4. Dimension.- x 5. The topology of a real finite dimensional vector space.- II. Linear mappings.- x 1. Basic properties.- x 2. Operations with linear mappings.- x 3. Linear isomorphisms.- x 4. Direct sum of vector spaces.- x 5. Dual vector spaces.- x 6. Finite dimensional vector spaces.- III. Matrices.- x 1. Matrices and systems of linear equations.- x 2. Multiplication of matrices.- x 3. Basis transformation.- x 4. Elementary transformations.- IV. Determinants.- x 1. Determinant functions.- x 2. The determinant of a linear transformation.- x 3. The determinant of a matrix.- x 4. Dual determinant functions.- x 5. The adjoint matrix.- x 6. The characteristic polynomial.- x 7. The trace.- x 8. Oriented vector spaces.- V. Algebras.- x 1. Basic properties.- x 2. Ideals.- x 3. Change of coefficient field of a vector space.- VI. Gradations and homology.- x 1. G-graded vector spaces.- x 2. G-graded algebras.- x 3. Differential spaces and differential algebras.- VII. Inner product spaces.- x 1. The inner product.- x 2. Orthonormal bases.- x 3. Normed determinant functions.- x 4. Duality in an inner product space.- x 5. Normed vector spaces.- x 6. The algebra of quaternions.- VIII. Linear mappings of inner product spaces.- x 1. The adjoint mapping.- x 2. Selfadjoint mappings.- x 3. Orthogonal projections.- x 4. Skew mappings.- x 5. Isometric mappings.- x 6. Rotations of Euclidean spaces of dimension 2, 3 and 4.- x 7. Differentiate families of linear automorphisms.- IX. Symmetric bilinear functions.- x 1. Bilinear and quadratic functions.- x 2. The decomposition of E.- x 3. Pairs of symmetric bilinear functions.- x 4. Pseudo-Euclidean spaces.- x 5. Linear mappings of Pseudo-Euclidean spaces.- X. Quadrics.- x 1. Affine spaces.- x 2. Quadrics in the affine space.- x 3. Affine equivalence of quadrics.- x 4. Quadrics in the Euclidean space.- XI. Unitary spaces.- x 1. Hermitian functions.- x 2. Unitary spaces.- x 3. Linear mappings of unitary spaces.- x 4. Unitary mappings of the complex plane.- x 5. Application to Lorentz-transformations.- XII. Polynomial algebra.- x 1. Basic properties.- x 2. Ideals and divisibility.- x 3. Factor algebras.- x 4. The structure of factor algebras.- XIII. Theory of a linear transformation.- x 1. Polynomials in a linear transformation.- x 2. Generalized eigenspaces.- x 3. Cyclic spaces.- x 4. Irreducible spaces.- x 5. Application of cyclic spaces.- x 6. Nilpotent and semisimple transformations.- x 7. Applications to inner product spaces.