2.20.3 Tensor Dual Basis Context

In tensor algebra, the dual basis provides a framework to pair tensors with their dual spaces, enabling coordinate-independent representations and transformations.

Tensor Dual Basis Context is the constructive machinery for producing, from any chosen basis of a vector space V, a uniquely determined companion basis of the dual space V*, together with the algorithms, uniqueness arguments, and coordinate-extraction techniques that make this pairing usable in tensor computation. Whereas the dual vector space context establishes the abstract relationship between V and V*, and the linear functional context studies a single covector in isolation, the dual basis context addresses the specific combinatorial problem: given n basis vectors of V, produce the n basis covectors of V* that satisfy the biorthogonality condition, and use them as coordinate-reading tools.

Existence and Uniqueness of the Dual Basis

The Defining Biorthogonality Relation

Given a basis e_1, ..., e_n of an n-dimensional vector space V, the dual basis e^1, ..., e^n of V* is the unique set of covectors satisfying

e^{i} (e_{j}) = δ_{j}^{i}

where δ^i_j is 1 when i = j and 0 otherwise. Existence follows because each e^i can be defined directly as the coordinate functional that extracts the i-th component of a vector expressed in the given basis; uniqueness follows because a linear functional on an n-dimensional space is completely determined by its values on the n basis vectors, and the biorthogonality relation prescribes those values exactly.

Dimension Matching

Since dim(V*) = dim(V) = n for finite-dimensional V, the n covectors e^1, ..., e^n constructed this way are automatically linearly independent and therefore form a full basis of V*, not merely a linearly independent subset. Linear independence follows directly from biorthogonality: if a combination c_i e^i vanished identically, evaluating it on each e_j would force every coefficient c_j to be zero.

Constructing the Dual Basis Explicitly

Matrix Inversion Method

If V = F^n with the standard pairing and the basis vectors e_1, ..., e_n are arranged as the columns of an invertible matrix M, then the dual basis covectors, written as row vectors, are exactly the rows of M^{-1}. This gives a direct computational recipe: to find the dual basis of any given basis, invert the matrix whose columns are the given basis vectors, and read off its rows.

M M^{- 1} = I

expresses biorthogonality directly, since the (i, j) entry of the product M^{-1}M is e^i(e_j), and this product equals the identity matrix precisely when the biorthogonality relations hold.

Worked Structure for a Non-Orthogonal Basis

For a basis that is not orthonormal, the dual basis covectors are generally not proportional to the original basis vectors, even after using an inner product to identify V with V*. Only for an orthonormal basis under a chosen inner product does the dual basis coincide, vector-for-vector, with the original basis; for a skewed or non-normalized basis, each e^i must lean away from the other basis vectors e_j with j ≠ i in order to annihilate them, which generally changes both its direction and its length relative to e_i.

Method for a General Inner Product Space

When V carries an inner product ⟨ , ⟩, an alternative construction identifies V* with V via the map v ↦ ⟨v, ·⟩, then finds the dual basis by solving the linear system ⟨g_i, e_j⟩ = δ_{ij} for vectors g_i in V; the resulting g_i are the images of e^i under this identification. This method reduces the dual basis problem to inverting the Gram matrix of the original basis, G_{ij} = ⟨e_i, e_j⟩, since the coefficients of each g_i in terms of e_1, ..., e_n are the entries of a row of G^{-1}.

Transformation of the Dual Basis Under Change of Basis

Contragredient Rule

If the basis of V changes via an invertible matrix A, so ẽ_j = A^i_j e_i, then to preserve biorthogonality the dual basis must transform by the inverse-transpose, ẽ^i = (A^{-1})^i_j e^j. This contragredient law guarantees that the biorthogonality relation ẽ^i(ẽ_j) = δ^i_j continues to hold after the change of basis, and it is the origin of the transformation rule applied to every lower tensor index.

Consistency Check via Composition

Composing the transformation of V's basis with the transformation of its dual basis and pairing the results reproduces the identity matrix, confirming internal consistency:

{\tilde{e}}^{i} ({\tilde{e}}_{j}) = {(A^{- 1})}_{k}^{i} A_{j}^{k} = δ_{j}^{i}

Using the Dual Basis to Extract Components

Coordinate Extraction Formula

The principal computational use of the dual basis is component extraction: for any vector v in V, the coefficient of e_i in the expansion v = v^i e_i is recovered simply by applying the corresponding dual basis covector, v^i = e^i(v). This turns the abstract existence of a basis expansion into an explicit, computable formula, avoiding the need to solve a linear system from scratch every time a vector's coordinates are needed.

Extraction of Tensor Components

The same mechanism generalizes directly to extracting the components of a general (p, q) tensor T: applying the dual basis covectors to the contravariant argument slots and the original basis vectors to the covariant argument slots isolates a single component,

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = T (e^{i_{1}}, \dots, e^{i_{p}}, e_{j_{1}}, \dots, e_{j_{q}})

reproducing the standard index notation for tensor components directly from the multilinear-map definition of a tensor.

Special Cases and Sanity Checks

The Standard Basis Is Self-Dual Under the Standard Pairing

For V = F^n with the standard basis and the standard pairing ⟨f, v⟩ = Σ f_i v^i, the dual basis coincides numerically with the standard basis itself, since the identity matrix is its own inverse. This special coincidence is often the source of the misconception that a basis and its dual are always "the same," a misconception that fails as soon as a non-orthonormal basis is used.

Reflexivity Consistency

Applying the dual basis construction twice, once to obtain e^i from e_i, and again to obtain the dual basis of V** from e^i, returns a basis of V** that matches e_i under the canonical identification V ≅ V**, confirming that the dual basis construction is compatible with double duality rather than an arbitrary or asymmetric procedure.