6.20.3 Tensor Covector Dual Basis Component Form

Tensor Covector Dual Basis Component Form is the expression of a covector as a linear combination of dual basis covectors weighted by its components, φ = φ_i e^i, providing the concrete, computable bridge between the abstract covector φ, an element of V* considered independently of any coordinate system, and the array of numbers φ_i that represent it once a specific dual basis {e^i} has been chosen. This form is the covariant counterpart to the vector's basis component form, and it is what allows abstract linear functionals to be added, scaled, and evaluated on vectors using ordinary arithmetic performed on their components.

Constructing the Dual Basis Component Form

Existence and Uniqueness of the Expansion

Given a basis {e_1, ..., e_n} of V and its associated dual basis {e^1, ..., e^n} of V*, defined by e^i(e_j) = δ^i_j, every covector φ in V* can be written uniquely as:

compressed using the summation convention into φ = φ_i e^i. Existence and uniqueness follow from the fact that the dual basis {e^i} is itself a basis of the n-dimensional space V*, exactly as {e_i} is a basis of V.

Extracting Components by Evaluation on Basis Vectors

The components of φ can be recovered directly from the abstract covector by evaluating it on the original basis vectors of V, φ_j = φ(e_j), which follows immediately from applying φ to e_j, and using e^i(e_j) = δ^i_j to collapse the sum in the dual basis component form to the single term φ_j.

Behavior of the Dual Basis Component Form Under Vector Space Operations

Addition and Scalar Multiplication

The dual basis component form respects the vector space operations of V* exactly: if φ = φ_i e^i and ψ = ψ_i e^i, then φ + ψ = (φ_i + ψ_i) e^i, and for any scalar c, cφ = (cφ_i) e^i. These identities justify performing the addition and scalar multiplication of linear functionals componentwise.

The Zero Covector in Dual Basis Component Form

The zero covector, the functional sending every vector to zero, corresponds to the dual basis component form with every component equal to zero, and because of the uniqueness of the expansion, a covector is zero if and only if all of its components relative to any dual basis are zero.

Changing the Dual Basis Component Form Under a Change of Basis

Recomputing Components in a New Dual Basis

If the basis of V changes via e'_i = A^k_i e_k, the dual basis transforms with the inverse matrix, e'^i = B^i_k e^k, and the same covector φ acquires new components φ'_i in the new expansion φ = φ'_i e'^i, related to the old components by the covariant transformation law φ'_i = A^k_i φ_k.

Consistency Check via Substitution

Substituting the new dual basis covectors and new components into the new expansion and simplifying using AB = I reproduces the original expansion φ = φ_k e^k exactly, confirming that the dual basis component form, despite using different numbers in different dual bases, always describes one and the same invariant covector.

Diagram of the Dual Basis Component Form

Relating the Dual Basis Component Form to the Vector Basis Component Form

Structural Parallel Between V and V*

The dual basis component form is the exact structural analogue of the ordinary vector basis component form, with V* and {e^i} playing the roles that V and {e_i} play for vectors; every statement made about existence, uniqueness, and change of basis for vector components has a direct counterpart for covector components, obtained by systematically exchanging the roles of A and B and of upper and lower indices.

Practical Use in Computing Pairings

Once both a vector and a covector are expressed in their respective basis component forms, the pairing φ(v) is computed directly as φ_i v^i, a simple sum of products of corresponding components, illustrating why the dual basis component form is the practical prerequisite for evaluating any linear functional numerically.