Elements of Linear Algebra

Abstract

In this chapter the fundamental concepts of linear algebra are exposed. More precisely, we introduce vector spaces, bases and contravariant components of a vector relative to a base, together with their transformation formulae on varying the basis. Then, Euclidean vector spaces and some fundamental operations in these spaces are analyzed: vector product, mixed product, etc. The elementary definition of n-tensors is given together with elements of tensor algebra. The problem of eigenvalues of symmetric 2-tensor and orthogonal 2-tensors is widely discussed and Cauchy’s polar decomposition theorem is proved. Finally, the Euclidean point spaces are introduced. The last sections contain exercises and detailed descriptions of some packages written with Mathematica^® [69], which allows the user to solve by computer many of the problems considered in this chapter.