14.5.2 Tensor Linear Functional Product Scalar Output

Tensor Linear Functional Product Scalar Output combines tensors and linear functionals to produce a scalar via structured algebraic operations.

Tensor Linear Functional Product Scalar Output is the single scalar value produced by evaluating the tensor product of two linear functionals on an elementary tensor, obtained by multiplying together the scalar output of each individual functional on its corresponding component.

Definition of the Scalar Output

The Product Formula

For linear functionals

φ : V \to F

ψ : W \to F

the scalar output on an elementary tensor v tensor w is

(φ \otimes ψ) (v \otimes w) = φ (v) \cdot ψ (w),

with the right side an ordinary product of two scalars in the field F, obtained after phi and psi have each already reduced their respective inputs to scalars.

Distinguishing the Scalar Output from a Tensor of Scalars

Although the general elementary output rule for tensor products of maps produces an elementary tensor phi(v) tensor psi(w) in F tensor F, the scalar output identifies this elementary tensor with the ordinary scalar product phi(v) psi(w) through the canonical isomorphism between F tensor F and F given by scalar multiplication, so the scalar output is properly understood as a single number in F rather than as a formal tensor of two numbers.

Extension to General Elements

Sums of Elementary Tensors

For a general element of V tensor W expressed as a finite sum of elementary tensors, the scalar output extends by linearity to

(φ \otimes ψ) (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} φ (v_{i}) \cdot ψ (w_{i}),

a single scalar obtained by summing the scalar outputs on each elementary tensor in the decomposition, and this sum is independent of which particular decomposition into elementary tensors is used, by the same well-definedness guaranteed for the general tensor product of maps.

Coordinate Computation of the Scalar Output

If v has coordinates c-1 through c-m with respect to a basis of V, and w has coordinates d-1 through d-n with respect to a basis of W, and if phi and psi have coordinate row vectors a and b in the corresponding dual bases, the scalar output reduces to

(φ \otimes ψ) (v \otimes w) = (\sum_{i} a_{i} c_{i}) \cdot (\sum_{j} b_{j} d_{j}),

an explicit product of two ordinary dot products, giving a directly computable formula for the scalar output once coordinates are known.

Properties of the Scalar Output

Bilinearity in the Functional Pair

The scalar output is bilinear in the pair phi and psi: fixing v and w, the map sending the pair of functionals to the scalar output is linear in phi and linear in psi separately, since

(φ_{1} + φ_{2}) (v) \cdot ψ (w) = φ_{1} (v) \cdot ψ (w) + φ_{2} (v) \cdot ψ (w) .

Zero Scalar Output

The scalar output vanishes on v tensor w whenever either phi vanishes at v or psi vanishes at w, so the kernel of phi tensor psi, viewed as a functional on V tensor W, contains every elementary tensor built from the kernel of phi paired with any w, or from any v paired with the kernel of psi, giving an explicit family of elementary tensors on which the scalar output is guaranteed to be zero without further computation.