3.7.5 Tensor Vector Covector Pairing Bilinearity

Tensor Vector Covector Pairing Bilinearity explores how tensors interact with vectors and covectors through bilinear mappings in multilinear algebra.

Tensor Vector Covector Pairing Bilinearity is the property that the natural pairing between V and V* is a bilinear form on the product space V x V*, meaning it is linear in the vector argument for every fixed covector and linear in the covector argument for every fixed vector, and this property qualifies the pairing as belonging to the broader class of bilinear forms studied throughout linear algebra, while also being distinguished among them by its nondegeneracy and its choice-free construction.

The Pairing as a Bilinear Form

General Definition of a Bilinear Form

A bilinear form on a pair of vector spaces V and W over a field F is a map B : V x W -> F that is linear in each argument separately. The vector-covector pairing is the specific bilinear form obtained by taking W = V* and defining B(v, f) = f(v). Unlike a general bilinear form, which could be any linear-in-each-slot map, this particular form is uniquely determined by the algebraic structure of V* as the dual of V, with no freedom to choose a different such form.

Verifying Bilinearity Directly

Linearity in v, for fixed f, follows because f is itself a linear functional:

B (a v_{1} + b v_{2}, f) = f (a v_{1} + b v_{2}) = a f (v_{1}) + b f (v_{2})

Linearity in f, for fixed v, follows because addition and scalar multiplication on V* are defined pointwise:

B (v, c f_{1} + d f_{2}) = (c f_{1} + d f_{2}) (v) = c f_{1} (v) + d f_{2} (v)

Together these two computations establish full bilinearity.

Matrix Representation of the Bilinear Form

The Identity Matrix as the Pairing's Matrix

Relative to a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n, the pairing bilinear form has matrix entries B(e_i, e^j) = e^j(e_i) = δ^j_i, so its matrix representation, when the dual basis is used on the covector side, is exactly the n x n identity matrix:

(\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix})

This is a direct consequence of how the dual basis is defined, and it is what makes the pairing computationally trivial once the dual basis is used, reducing to the sum of matching-index products, v^i f_i.

Why the Matrix Is the Identity Only in Dual Bases

If a covector f were instead expressed in components relative to some other basis of V* not dual to the chosen basis of V, the matrix representing B would not be the identity matrix; it would instead encode the relationship between the two independently chosen bases. The identity-matrix form is special to the pairing of a basis with its own dual basis.

Nondegeneracy as a Bilinearity Consequence

Definition of Nondegenerate

A bilinear form B : V x W -> F is called nondegenerate if the only v in V with B(v, w) = 0 for all w in W is v = 0, and symmetrically for W. The vector-covector pairing is nondegenerate on both sides: if f(v) = 0 for every covector f, then v = 0, since the dual basis covectors alone would otherwise detect a nonzero component of v.

Nondegeneracy Distinguishes the Pairing from Arbitrary Bilinear Forms

Not every bilinear form on a pair of spaces is nondegenerate; a bilinear form can easily be constructed with a nontrivial set of vectors that pair to zero with everything in the other space. The natural pairing between V and V* is always nondegenerate precisely because V* is defined as the full space of all linear functionals on V, leaving no vector able to hide from every possible functional.

Diagrammatic Summary

The diagram summarizes the two independent linearity properties that combine to make the pairing a bilinear form, together with its defining nondegeneracy.