## 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. |

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### Contents

nDimensional Spaces Linear and Bilinear Forms | 1 |

Linear Transformations | 70 |

The Canonical Form of an Arbitrary Linear Transformation | 132 |

Introduction to Tensors | 164 |

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### Common terms and phrases

algebra arbitrary 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 Dk(X dual eigenvalues eigenvector elementary divisors elementary transformations elements equal equivalent Euclidean space Example EXERCISE exists follows geometry Gramm determinant Hence Hermitian quadratic form implies inner product invariant subspace inverse invertible matrix isomorphism Jordan canonical form Lemma linear combination linear function linear transformation linearly independent linearly independent vectors MATHEMATICS matrix sf 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 three•dimensional tion two•dimensional unitary transformation