Linear Algebra and Multidimensional Geometry
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TOC
CHAPTER I. LINEAR VECTOR SPACES AND LINEAR MAPPINGS
1. The sets and mappings
2. Linear vector spaces
3. Linear dependence and linear independence
4. Spanning systems and bases
5. Coordinates. Transformation of the coordinates of a vector under a change of basis
6. Intersections and sums of subspaces
7. Cosets of a subspace. The concept of factorspace
8. Linear mappings
9. The matrix of a linear mapping
10. Algebraic operations with mappings
The space of homomorphisms Hom(V;W)
CHAPTER II. LINEAR OPERATORS
1. Linear operators. The algebra of endomorphisms End(V ) and the group of automorphisms Aut(V )
2. Projection operators
3. Invariant subspaces. Restriction and factorization of operators
4. Eigenvalues and eigenvectors
5. Nilpotent operators
6. Root subspaces. Two theorems on the sum of root subspaces
7. Jordan basis of a linear operator. Hamilton-Cayley theorem
CHAPTER III. DUAL SPACE
1. Linear functionals. Vectors and covectors. Dual space
2. Transformation of the coordinates of a covector under a change of basis
3. Orthogonal complements in a dual spaces
4. Conjugate mapping
CHAPTER IV. BILINEAR AND QUADRATIC FORMS
1. Symmetric bilinear forms and quadratic forms. Recovery formula
2. Orthogonal complements with respect to a quadratic form
3. Transformation of a quadratic form to its canonic form Inertia indices and signature
4. Positive quadratic forms. Silvester’s criterion
CHAPTER V. EUCLIDEAN SPACES
1. The norm and the scalar product. The angle between vectors Orthonormal bases
2. Quadratic forms in a Euclidean space. Diagonalization of a pair of quadratic forms
3. Selfadjoint operators. Theorem on the spectrum and the basis of eigenvectors for a selfadjoint operator
4. Isometries and orthogonal operators
CHAPTER VI. AFFINE SPACES
1. Points and parallel translations. Ane spaces
2. Euclidean point spaces. Quadrics in a Euclidean space
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