14.3.3 Tensor Map Product Output Assignment

Tensor Map Product Output Assignment defines how tensor maps interact, determining the structure and properties of their resulting tensor products in algebraic contexts.

Tensor Map Product Output Assignment is the rule that specifies, for each element of the domain tensor product, exactly which element of the codomain tensor product a tensor product of maps assigns to it, distinguishing the assignment on elementary tensors from its extension to arbitrary elements.

Assignment on Elementary Tensors

The Elementary Output Rule

For linear maps f and g forming the factor map pair, the output assignment on an elementary tensor v tensor w is

(f \otimes g) (v \otimes w) = f (v) \otimes g (w),

assigning to each elementary tensor the elementary tensor formed from the individual outputs of f and g. This is the atomic case of the assignment, from which every other case is derived.

Well-Definedness of the Elementary Assignment

Because a single element of the tensor product may be an elementary tensor of more than one pair of vectors when relations hold among the vectors involved, the output assignment on elementary tensors must be checked to give the same output regardless of which representative pair is used. This is guaranteed because the elementary output rule arises from a bilinear map, and bilinearity automatically respects all such relations among elementary tensors.

Assignment on General Elements

Linear Extension of the Output

For a general element expressed as a finite sum of elementary tensors, the output assignment extends by linearity,

(f \otimes g) (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} f (v_{i}) \otimes g (w_{i}),

so that the output on any element is obtained by applying the elementary output rule term by term and adding the resulting elementary tensors in the codomain.

Independence from the Chosen Decomposition

Since a general element admits multiple decompositions into sums of elementary tensors, the output assignment is meaningful only because it produces the same result under every valid decomposition, a fact secured by the universal property used to construct f tensor g rather than by any direct verification on each decomposition individually.

Assignment Under Special Configurations

Assignment When One Factor Is Fixed

When the output assignment is restricted to elements of the form v tensor w for a single fixed w, varying only v, the assignment reduces to

v \mapsto f (v) \otimes g (w),

a linear assignment in v alone, scaled by the fixed vector g of w, illustrating how the joint output assignment decomposes into a family of simpler assignments once one argument is held constant.

Assignment for the Zero Element

The output assignment necessarily sends the zero element of the domain tensor product to the zero element of the codomain tensor product, consistent with linearity, and this holds regardless of how the zero element happens to be expressed as a sum of elementary tensors, since every valid decomposition of zero must produce a sum of individual outputs that itself cancels to zero.

Consequences for Image and Preimage

Determining the Image

The output assignment, applied to a spanning set of elementary tensors of the domain, generates a spanning set for the image of f tensor g, so the image is completely determined by evaluating the output assignment on elementary tensors built from bases of the two domain factors, without needing to consider every element of the domain individually.

Determining Preimages

Finding a preimage of a given element under the output assignment reduces, via the Kronecker product representation, to solving a linear system whose coefficient matrix is the Kronecker product of the matrices of f and g, so that questions about which elements of the codomain are reached by the output assignment can be answered by standard linear-system techniques applied to this explicit matrix.