# Book:I.M. Gel'fand/Lectures on Linear Algebra/Second Edition

## I.M. Gel'fand: Lectures on Linear Algebra (2nd Edition)

Published $\text {1961}$, Dover Publications

ISBN 0-486-66082-6 (translated by A. Shenitzer)

### Contents

Preface to the Second Edition (September 1950)
Preface to the First Edition (January 1948)

$\text {I}$. $n$-Dimensional Spaces. Linear and Bilinear Forms
$\S 1$. $n$-Dimensional vector spaces
$\S 2$. Euclidean space
$\S 3$. Orthogonal basis. Isomorphism of Euclidean spaces
$\S 4$. Bilinear and quadratic forms
$\S 5$. Reduction of a quadratic form to a sum of squares
$\S 6$. Reduction of a quadratic form by means of a triangular transformation
$\S 7$. The law of inertia
$\S 8$. Complex $n$-dimensional space
$\text {II}$. Linear Transformations
$\S 9$. Linear transformations. Operations on linear transformations
$\S 10$. Invariant subspaces. Eigenvalues and eigenvectors of a linear transformation
$\S 11$. The adjoint of a linear transformation
$\S 12$. Self-adjoint (Hermitian) transformations. Simultaneous reduction of a pair of quadratic forms to a sum of squares
$\S 13$. Unitary transformations
$\S 14$. Commutative linear transformations. Normal transformations
$\S 15$. Decomposition of a linear transformation into a product of a unitary and self-adjoint transformation
$\S 16$. Linear transformations on a real Euclidean space
$\S 17$. Extremal properties of eigenvalues
$\text {III}$. The Canonical Form of an Arbitrary Linear Transformation
$\S 18$. The canonical form of a linear transformation
$\S 19$. Reduction to canonical form
$\S 20$. Elementary divisors
$\S 21$. Polynomial matrices
$\text {IV}$. Introduction to Tensors
$\S 22$. The dual space
$\S 23$. Tensors

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