3.7.1 Tensor Vector Covector Input Pair

A Tensor Vector Covector Input Pair combines a vector and covector to form a tensor, enabling linear operations in multilinear algebra.

Tensor Vector Covector Input Pair is the ordered pair (v, f), drawn from the Cartesian product V x V*, that serves as the single argument accepted by the vector-covector pairing operation, and understanding its structure clarifies exactly what data the pairing consumes before it produces a scalar. The pair consists of one vector v chosen from V and one covector f chosen from the dual space V*, and the pairing operation is defined on the entire product space V x V*, treating the pair as one composite input rather than two independent ones supplied separately.

Structure of the Input Pair

The Cartesian Product Domain

The domain of the pairing operation is the set

V \times V^{*} = ((v, f) : v \in V, f \in V^{*})

Every element of this set is an ordered pair: the first component always belongs to V, and the second always belongs to V*. Two pairs (v_1, f_1) and (v_2, f_2) are equal precisely when v_1 = v_2 and f_1 = f_2, so the pair genuinely encodes two pieces of independent information bundled together.

Why Order Matters in the Notation, Not in the Value

Writing the input as (v, f) fixes a convention for which slot is the vector and which is the covector, distinguishing it from a pair (f, v) drawn from V* x V. The two conventions describe the same mathematical content and, once paired through the evaluation map, produce the same scalar f(v), but as formal ordered pairs from different Cartesian products, (v, f) and (f, v) are technically distinct objects living in different domains.

The Input Pair as a Vector Space Element

Direct Sum Structure

The product V x V* can itself be equipped with a vector space structure, with addition and scalar multiplication defined componentwise:

(v_{1}, f_{1}) + (v_{2}, f_{2}) = (v_{1} + v_{2}, f_{1} + f_{2})

This makes V x V* the external direct sum of V and V*, a vector space of dimension 2n when V has dimension n. It is important to note, however, that the pairing operation itself is not linear on this direct sum space; it is only bilinear when the vector part and the covector part are varied one at a time.

Distinguishing the Input Pair from the Tensor Product

The Cartesian product V x V*, whose elements are the input pairs, should not be confused with the tensor product space V ⊗ V*, whose elements are the (1, 1) tensors themselves. Every input pair (v, f) determines a specific element v ⊗ f of the tensor product, but not every element of V ⊗ V* arises from a single pair; general elements of the tensor product are sums of several such simple pairs.

From Input Pair to Output Scalar

The Evaluation Map on Pairs

The pairing operation is the function

E : V \times V^{*} \to F, E (v, f) = f (v)

Every input pair maps to exactly one scalar output, and every scalar in F is attainable as such an output whenever V is nonzero, since for any nonzero v a covector can always be constructed sending v to any desired scalar.

Component Description of the Pair

Relative to a basis, the input pair (v, f) is fully specified by 2n numbers: the n contravariant components v^1, ..., v^n of the vector and the n covariant components f_1, ..., f_n of the covector. The pairing operation then reduces these 2n numbers to the single scalar v^i f_i through the summation formula.

Diagrammatic Summary

The diagram represents the ordered pair (v, f) as a single composite input drawn from V x V*, which the evaluation map E reduces to one scalar output.