3.6.4 Tensor Covector Evaluation Component Form

Tensor covector evaluation in component form assesses how covectors act on vectors through indexed tensor components in multilinear algebra.

Tensor Covector Evaluation Component Form is the coordinate expression of the covector-vector pairing as a sum of products of matching components, written f_i v^i under the Einstein summation convention, and it is the practical computational tool that replaces the abstract evaluation f(v) with an explicit arithmetic procedure once a basis has been chosen. This component form is derived directly from the bilinearity of the evaluation operation together with the defining relation between a basis and its dual basis, and it is the formula actually used whenever a covector-vector pairing is computed by hand or by machine.

Deriving the Component Formula

Setting Up a Basis and Dual Basis

Let e_1, ..., e_n be a basis of V, and let e^1, ..., e^n be the corresponding dual basis of V*, uniquely determined by the requirement

e^{i} (e_{j}) = δ_{j}^{i}

where δ^i_j is the Kronecker delta, equal to 1 when i = j and 0 otherwise. Any vector v in V can be written as v = v^i e_i, and any covector f in V* can be written as f = f_i e^i, where v^i and f_i are the respective components relative to these bases.

Expanding the Pairing

Applying the evaluation operation to f and v, and using bilinearity to distribute the sums,

f (v) = (f_{i} e^{i}) (v^{j} e_{j}) = f_{i} v^{j} e^{i} (e_{j})

Substituting e^i(e_j) = δ^i_j collapses the double sum to a single sum, since the Kronecker delta forces i = j:

f (v) = f_{i} v^{j} δ_{j}^{i} = f_{i} v^{i}

This last expression, f_i v^i, is the component form of the evaluation operation.

The Einstein Summation Convention

Repeated Indices Imply Summation

Under the Einstein summation convention, whenever an index appears exactly once as a subscript and once as a superscript within a single term, summation over that index from 1 to n is implied without writing an explicit summation sign. The expression f_i v^i therefore stands for

f_{i} v^{i} = \sum_{i = 1}^{n} f_{i} v^{i}

Why Only Upper-Lower Pairs Are Summed

The convention restricts summation to index pairs where one occurrence is upper and the other is lower, because only such pairs correspond to a genuine contraction between a contravariant slot and a covariant slot. An expression like f_i v_i, with both indices lower, would not represent a basis-independent operation, since covector components and vector components transform oppositely under a change of basis; only the mixed pairing f_i v^i produces a quantity invariant under such changes.

Concrete Numerical Example

A Worked Computation in Three Dimensions

Suppose V = R^3 with the standard basis, and suppose a covector has components f = (2, -1, 4) while a vector has components v = (3, 0, 5). The component form gives

f (v) = f_{1} v^{1} + f_{2} v^{2} + f_{3} v^{3}

= (2 \times 3) + (- 1 \times 0) + (4 \times 5) = 6 + 0 + 20 = 26

This matches the familiar dot-product calculation used in elementary linear algebra, showing that the dot product on R^n is a special case of the covector-vector evaluation once covectors are identified with row vectors, or equivalently with ordinary vectors via a fixed inner product.

Matrix Representation

Row Times Column

If the components f_i are arranged as a row array and the components v^i are arranged as a column array, the component formula f_i v^i is exactly the matrix product of a 1 x n row matrix with an n x 1 column matrix, producing a 1 x 1 matrix, that is, a scalar:

f (v) = (\begin{matrix} f_{1} & f_{2} & \dots & f_{n} \end{matrix}) (\begin{matrix} v^{1} \\ v^{2} \\ ⋮ \\ v^{n} \end{matrix})

This matrix picture is why covectors are frequently represented as row vectors and vectors as column vectors, since the evaluation operation then coincides exactly with ordinary matrix multiplication.

Diagrammatic Summary

The diagram shows the two component arrays being multiplied entry by entry and then summed, illustrating the mechanical procedure that the abstract formula f_i v^i describes.