2.10 Tensor Coordinate Vector Representation

Tensor Coordinate Vector Representation is the practice of describing a vector from a vector space used in tensor construction by the ordered tuple of coefficients obtained when that vector is expressed as a linear combination of a fixed basis, allowing abstract vectors to be manipulated as concrete lists of numbers. This representation is the primary bridge between the abstract algebra of vector spaces and the numerical computations required to work with tensors in practice.

Formal Statement

Definition of the Coordinate Vector

Given a fixed basis of a vector space and a vector expressed as a linear combination of that basis, the coordinate vector is the tuple formed from the coefficients of that linear combination, taken in the order the basis vectors are listed.

Correspondence Between Vectors and Tuples

Coordinate vector representation establishes a correspondence in which each vector maps to exactly one coordinate tuple relative to a given ordered basis, and conversely each coordinate tuple maps back to exactly one vector.

Requirements Enabling the Representation

Need for a Fixed, Ordered Basis

Coordinate vector representation requires that the basis be treated as an ordered list rather than a mere set, since the position of each coefficient in the tuple corresponds to a specific basis vector, and reordering the basis reorders the coordinate tuple accordingly.

Reliance on Coverage and Independence

The representation is well defined only because the basis both spans the space, guaranteeing every vector has at least one coordinate tuple, and is linearly independent, guaranteeing that tuple is unique.

Operations Reflected in Coordinates

Addition of Vectors as Addition of Tuples

Adding two vectors corresponds exactly to adding their coordinate tuples entrywise, since the coefficients of a sum of linear combinations are the sums of the corresponding coefficients.

Scalar Multiplication as Entrywise Scaling

Scaling a vector by a field element corresponds to scaling every entry of its coordinate tuple by that same element, preserving the linear structure of the vector space in coordinate form.

Role in Tensor Construction

Enabling Numerical Tensor Components

Coordinate vector representation is what allows the components of a tensor built from several vector spaces to be described as explicit numerical arrays, since each factor vector contributing to the tensor is first reduced to its coordinate tuple.

Sensitivity to Basis Choice

Because coordinate vectors are always defined relative to a specific basis, the actual numbers appearing in a coordinate vector change when the basis changes, even though the underlying vector they describe does not, a fact made precise by basis change formulas.

Summary of Key Properties

Concrete Stand-In for Abstract Vectors

Tensor Coordinate Vector Representation gives every vector a concrete, ordered list of numbers that faithfully stands in for the vector once a basis is fixed, enabling direct computation.

Foundation for Component-Based Tensor Algebra

This representation underlies component list structure, basis dependence, and coordinate ordering, all of which refine how coordinate vectors are organized and interpreted within tensor algebra.