## Lectures on Linear AlgebraProminent Russian mathematician's concise, well-written exposition considers n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

nDimensional Spaces Linear and Bilinear Forms | 1 |

Euclidean space | 14 |

Orthogonal basis Isomorphism of Euclidean spaces | 21 |

Bilinear and quadratic forms | 34 |

Reduction of a quadratic form to a sum of squares | 42 |

Reduction of a quadratic form by means of a triangular trans formation | 46 |

The law of inertia | 55 |

Complex ndimensional space | 60 |

Unitary transformations | 103 |

Commutative linear transformations Normal transformations | 107 |

Decomposition of a linear transformation into a product of a unitary and selfadjoint transformation | 111 |

Linear transformations on a real Euclidean space | 114 |

Extremal properties of eigenvalues | 126 |

The Canonical Form of an Arbitrary Linear Transformation | 132 |

Reduction to canonical form | 137 |

Elementary divisors | 142 |

Linear Transformations | 70 |

Invariant subspaces Eigenvalues and eigenvectors of a linear transformation | 81 |

The adjoint of a linear transformation | 90 |

Selfadjoint Hermitian transformations Simultaneous reduc | 97 |

tion of a pair of quadratic forms to a sum of squares | 100 |

Polynomial matrices | 149 |

Introduction to Tensors | 164 |

Tensors | 171 |

### Other editions - View all

### Common terms and phrases

arbitrary basis e1 basis relative basis vectors change of basis characteristic polynomial choice of basis coefficients column complex numbers compute corresponding defined denote different from zero dimension dimensional dual eigenvalues eigenvector elementary divisors elementary transformations elements equal Euclidean space Example Exercises exists follows geometry Hence Hermitian quadratic form implies inner product invariant subspace inverse isomorphism Jordan canonical form kth order minors Lemma linear combination linear function linear transformation linearly independent linearly independent vectors mathematics multilinear function multiplication n-dimensional Euclidean space n-dimensional space n-dimensional vector space n-tuples necessary and sufficient non-singular non-zero number of linearly obtain orthogonal basis orthogonal transformation orthonormal basis polynomial matrix polynomials of degree positive definite problems proof prove quadratic form real numbers scalar self-adjoint transformation set of vectors sum of squares tensor of rank theorem theory tion unitary transformation vectors e1