3.7.2 Tensor Vector Covector Pairing Scalar Result

The pairing of a tensor, vector, and covector yields a scalar, revealing deep algebraic relationships in multilinear algebra.

Tensor Vector Covector Pairing Scalar Result is the field element produced when the vector-covector pairing operation is applied to an input pair (v, f), and examining its properties clarifies what kind of object emerges from the pairing and how that output behaves under further algebraic manipulation. The result, written <v, f> or equivalently f(v), is always a single number in the field F underlying the vector space V, never a vector, never a covector, and never a higher-rank tensor, regardless of the dimension of V or the particular choice of v and f.

Characterizing the Result

Field Membership

For any pair (v, f) in V x V*, the scalar result <v, f> lies in F. If V is a real vector space, the result is a real number; if V is a complex vector space, the result is a complex number. The result carries no leftover geometric structure from V or V*, since all index structure is consumed in the pairing.

Rank Zero

Expressed in index notation as v^i f_i, the summed quantity has no free upper or lower indices remaining, since the single upper index on v^i is matched and summed against the single lower index on f_i. An object with no free indices is a (0, 0) tensor, which is precisely the technical definition of a scalar within tensor algebra.

Dependence of the Result on Its Inputs

Bilinear Dependence

The scalar result depends on both v and f bilinearly. Fixing f and varying v linearly produces a linear map V -> F, which is exactly the functional f itself, recovered from the pairing. Fixing v and varying f linearly produces a linear map V* -> F, which is exactly the functional ι(v) on V* corresponding to v under the canonical double-dual embedding.

Zero Results

The scalar result is zero precisely when v lies in the kernel of f, that is, when f(v) = 0. For a fixed nonzero covector f, the set of vectors v producing a zero result forms a hyperplane in V, a subspace of dimension n - 1 when V has dimension n, called the kernel or null space of f.

ker (f) = (v \in V : f (v) = 0)

Nondegeneracy of the Pairing

The pairing between V and V* is nondegenerate: if f(v) = 0 for every f in V*, then v must be the zero vector, and if f(v) = 0 for every v in V, then f must be the zero covector. This nondegeneracy is what allows the pairing to fully distinguish distinct vectors using covectors, and distinct covectors using vectors.

Algebraic Properties of the Scalar Result

Behavior Under Addition of Inputs

Since the scalar result is bilinear, changing either input by addition changes the result additively:

〈 v_{1} + v_{2}, f 〉 = 〈 v_{1}, f 〉 + 〈 v_{2}, f 〉

Behavior Under Scaling

Scaling either input by a scalar c scales the result by the same amount:

〈 c v, f 〉 = c 〈 v, f 〉 = 〈 v, c f 〉

Scaling both inputs simultaneously by c scales the result by c^2, since the bilinearity applies to each factor independently.

Uses of the Scalar Result

Component Extraction

When f is chosen to be a dual basis covector e^i, the scalar result e^i(v) extracts the i-th coordinate of v directly, making the pairing the fundamental mechanism for reading off numerical components from an abstract vector.

Building Higher Invariants

Scalar results from repeated pairings are combined to build invariant quantities used throughout physics and geometry, such as the trace of a linear operator, obtained by summing the scalar results e^i(T e_i) over a basis, or the norm of a vector under a chosen inner product, obtained by pairing a vector with its image under a metric-induced covector.

Diagrammatic Summary

The diagram summarizes how the scalar result responds linearly to changes in each input separately, and identifies the special case in which the result vanishes.