On some problems of tensor calculus. II

Basic definitions of linear algebra and functional analysis are given. In particular, the definitions of a semigroup, group, ring, field, module, and linear space are given 1–3, 6]. A local theorem on the existence of homeomorphisms is stated. Definitions of the inner r-product, local inner product of tensors whose rank is not less than r, and of local norm of a tensor 22] are also given. Definitions are given and basic theorems and propositions are stated and proved concerning the linear dependence and independence of a system of tensors of any rank. Moreover, definitions and proofs of some theorems connected with orthogonal and biorthonormal tensor systems are given. The definition of a multiplicative basis (multibasis) is given and ways of construction bases of modules using bases of modules of smaller dimensions. In this connection, several theorems are stated and proved. Tensor modules of even orders and problems on finding eigenvalues and eigentensors of any even rank are studied in more detail than in 22]. Canonical representations of a tensor of any even rank are given. It is worth while to note that it was studied by the Soviet scientist I. N. Vekua, and an analogous problem for the elasticity modulus tensor was considered by the Polish scientist Ya. Rikhlevskii in 1983–1984.