Preamble

The functions of space and time (i.e., fields) used to model material behaviour take values which may be real numbers, vectors, or higher-order tensors. Formal manipulations of tensors (i.e., tensor algebra) are best understood in terms of vector spaces. Here basic concepts and results are reviewed for completeness and for establishing familiarity with the notation employed. Vectorial entities (i.e., entities which have both direction and magnitude and combine like displacements) are modelled in terms of a three-dimensional inner-product vector space V, and higher-order tensorial entities are described in terms of algebraic constructs of V.

Simple considerations of rectilinear changes of position (i.e., displacements) and the notion of perpendicularity are used to establish the three-dimensional inner product vector space V used to model vectorial quantities, irrespective of their physical dimensions of mass, length, and time, and units of measurement. Linear transformations on V are defined and shown to have algebraic features in common with V, so motivating the definition of a general abstract vector space. The transpose of a linear transformation L on V and the tensor product of two vectors are defined without recourse to basis-dependent representations: such representations are derived upon selecting an orthonormal basis for V. Criteria which establish the invertibility or otherwise of a linear transformation L on V are identified, and the principal invariants and characteristic equation of L are analysed using alternating trilinear forms on V.