Lectures on Linear Algebra

I.    N-dimensional spaces. Linear and bilinear forms......................1


§ 1.  N-dimensional vector spaces..........................................1
§ 2.  Euclidean space.....................................................14
§ 3.  Orthogonal basis. Isomorphism of euclidean spaces...................21
§ 4.  Bilinear and quadratic forms........................................34
§ 6.  Reduction of a quadratic form to a sum of squares...................42
§ 6.  Reduction of a quadratic form by means of a
      triangular transformation...........................................46
§ 7.  The law of inertia..................................................55
§ 8.  Complex n-dimensional space.........................................60



II.   Linear transformations..............................................70


§  9. Linear transformations. Operations on linear transformations........70
§ 10. Invariant subspaces. Eigenvalues and eigenvectors of a 
      linear transformation...............................................81
§ 11. The adjoint of a linear transformation..............................90
§ 12. Self-adjoint (hermitian) transformations. Simultaneous reduction
      of a pair of quadratic forms to a sum of squares....................97
§ 13. Unitary transformations............................................108
§ 14. Commutative linear transformations. Normal transformations.........107
§ 15. Decomposition of a linear transformation into a product of
      a unitary and self-adjoint transformation..........................111
§ 16. Linear transformations on a real euclidean space...................114
§ 17. Extremal properties of eigenvalues.................................126



III.  The canonical form of an arbitrary linear transformation...........132

§ 18. The canonical form of a linear transformation......................132
§ 19. Reduction to canonical form........................................137
§ 20. Elementary divisors................................................142
§ 21. Polynomial matrices................................................149



IV.   Introduction to tensors............................................164

§ 22. The dual space.....................................................164
§ 23. Tensors............................................................171