2.15.1 Tensor Finite Basis Context

In tensor algebra, a finite basis context provides a structured framework for representing and manipulating tensors through a finite set of basis elements.

Tensor Finite Basis Context is the working assumption that the finite-dimensional vector space V underlying a tensor construction has been equipped with a fixed, finite, ordered basis e_1, ⋯, e_n, together with its associated dual basis e^1, ⋯, e^n of V*. Where the finite-dimensional context guarantees only that such a basis could exist, the finite basis context is the sharper working setting in which one has actually been chosen and fixed, so that vectors, covectors, and tensors can be written down as explicit, finite arrays of numbers indexed against that basis.

The Basis and Its Dual

Choosing a Basis for V

A finite basis e_1, ⋯, e_n for V is a linearly independent spanning set, so that every vector v in V has a unique expression

v = \sum_{i = 1}^{n} v^{i} e_{i}

where the scalars v^1, ⋯, v^n are the components of v relative to that basis. Fixing the basis turns the abstract vector v into a concrete finite list of n numbers.

The Dual Basis

Once e_1, ⋯, e_n is fixed, there is a uniquely determined dual basis e^1, ⋯, e^n for V*, characterized by the biorthogonality relation

e^{i} (e_{j}) = δ_{j}^{i}

where δ^i_j is the Kronecker delta, equal to 1 when i = j and 0 otherwise. Each dual basis covector e^i acts on a vector by extracting its i-th component, e^i(v) = v^i.

Index Ranges and Finite Summation

Bounded Index Sets

Because the basis is finite, every index attached to a tensor component ranges over the finite set {1, 2, ⋯, n}. This bounded range is what makes the Einstein summation convention meaningful: an expression such as v^i e_i denotes a finite sum over i = 1 to n, with no question of convergence, since only finitely many nonzero terms are ever summed.

Component Arrays as Finite Tables

A type (p, q) tensor, once the basis is fixed, is represented by a finite table of components T^{i_1 ⋯ i_p}_{j_1 ⋯ j_q}, with each of the p + q indices ranging independently over 1 through n. The total number of entries in this table is n^(p+q), matching the dimension of the tensor space, and every entry is an ordinary scalar that can be written down and computed with directly.

Change of Basis

The Transition Matrix

If a second basis f_1, ⋯, f_n is introduced, it is related to the first by an invertible transition matrix A, with

f_{j} = \sum_{i = 1}^{n} A_{j}^{i} e_{i}

Because both bases are finite, A is an ordinary n × n matrix with an ordinary matrix inverse, and this matrix, together with its inverse, is exactly what appears in the transformation law for tensor components between the two bases.

Component Transformation as Finite Matrix Arithmetic

Under the change of basis, the components of a vector transform as v'^i = (A^{-1})^i_j v^j, and more generally, the components of a type (p, q) tensor transform by applying A once for each contravariant index and A^{-1} once for each covariant index. Because n is finite and A is a finite matrix, every such transformation reduces to finite matrix multiplication, which is what allows tensor components to be computed numerically in any concrete basis, such as the standard basis of F^n.

Coordinate-Free Objects Versus Coordinate Representations

The Object Behind the Table

The finite basis context makes it possible to represent a tensor by a component table, but the tensor itself, as a coordinate-free object of T^p_q(V), does not depend on the basis chosen. Two different finite bases produce two different component tables for the same tensor, related by the transformation law; neither table is more "correct" than the other.

Practical Role of a Fixed Basis

Fixing a basis is nonetheless essential for computation: matrix representations of linear maps, explicit formulas for contraction, and numerical evaluation of tensor expressions all require an actual finite list of basis vectors against which components can be read off. The finite basis context is precisely the setting in which this reading-off is always possible and always produces a finite, well-defined array.

Diagrammatic Summary

The diagram shows the finite basis of V paired with its dual basis of V* through the biorthogonality relation, with both index sets bounded by the finite dimension n, which is the essential content of the finite basis context.