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

Subject Matter

 * Linear Algebra

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



Source work progress
* : $\S 1$: $n$-Dimensional vector spaces: Definition $1$