3.9.3 Tensor Dual Coordinate Component List

Tensor Dual Coordinate Component List is the ordered sequence of n scalars (f_1, f_2, ..., f_n) produced by the dual coordinate assignment, treated here as a concrete data object in its own right, together with the rules for how such lists are indexed, compared, and combined. While earlier discussion of dual coordinates emphasizes the assignment procedure and its basis dependence, the component list itself is the finished numerical artifact that calculations, storage, and comparison actually operate on.

Structure of the Component List

Indexing Convention

The component list of a covector f relative to a basis e_1, ..., e_n is indexed from 1 to n, with the i-th entry equal to f_i = f(e_i). The list is ordered, meaning the position of each entry matters: the list (2, -1, 0) is a different list from (-1, 2, 0), even though both contain the same multiset of numbers, because they correspond to different assignments of value to basis vector.

Length Equals Dimension

The length of the component list always equals n, the dimension of V, since the dual basis e^1, ..., e^n used to build the list has exactly as many elements as the original basis of V. A covector defined on a 5-dimensional space always has a component list of exactly 5 entries, never more, never fewer.

Equality and Comparison of Component Lists

Equality Criterion

Two component lists (f_1, ..., f_n) and (g_1, ..., g_n), both taken relative to the same basis, describe the same covector if and only if they are entrywise equal, f_i = g_i for every i from 1 to n. This criterion follows directly from the fact that a linear functional is uniquely determined by its values on a basis, so identical values on every basis vector force the two functionals to coincide everywhere.

Comparison Across Different Bases

Two component lists taken relative to different bases cannot be compared directly by checking entrywise equality; they must first be converted to a common basis using the covariant transformation rule, since the same covector generally produces different-looking, non-identical lists in different bases.

Arithmetic on Component Lists

Addition of Component Lists

Adding two covectors corresponds exactly to adding their component lists entrywise, since covector addition is defined pointwise: if f has component list (f_1, ..., f_n) and g has component list (g_1, ..., g_n), then f + g has component list (f_1 + g_1, ..., f_n + g_n).

Scalar Multiplication of Component Lists

Scaling a covector by c corresponds to scaling every entry of its component list by c: the covector cf has component list (c f_1, c f_2, ..., c f_n). Together with entrywise addition, this confirms that the set of all component lists, F^n, forms a vector space isomorphic to V* under the coordinate assignment, with the vector space operations matching exactly.

The Component List as a Row Matrix

Matrix Representation

The component list is frequently displayed as a 1 x n row matrix,

which allows the evaluation of the covector on a vector, whose components are arranged as an n x 1 column matrix, to be computed as an ordinary matrix product, producing a 1 x 1 matrix, that is, a single scalar.

List Concatenation Under Direct Sums

If V decomposes as a direct sum V = V_1 ⊕ V_2, a covector on V restricts to covectors on V_1 and V_2, and its component list, relative to a basis built by concatenating bases of V_1 and V_2, similarly splits into the concatenation of the component lists of the two restrictions.

Practical Storage and Manipulation

Representing Component Lists Computationally

In numerical or symbolic computation, a component list is naturally stored as a one-dimensional array of length n, and the operations of addition, scaling, and pairing with a vector's component list correspond directly to elementwise array addition, elementwise array scaling, and a dot-product-style reduction, respectively.

Order Sensitivity in Implementation

Because the position of each entry in the list corresponds to a specific basis vector, any implementation manipulating component lists must keep the ordering of entries consistent with a single, fixed enumeration of the basis; reordering the list without correspondingly reordering the basis produces a list describing a different covector.

Diagrammatic Summary

The diagram shows the ordered slots of a component list, each entry tied to the corresponding basis vector, emphasizing that position and meaning are inseparable in this representation.