14.8 Tensor Map Product Codomain Structure

Tensor Map Product Codomain Structure is the organization of everything specific to the target side of a tensor product of maps, encompassing the two codomain factor spaces, the tensor product they form together, and the way elements of that tensor product serve as the outputs produced by the induced map.

Constituents of the Codomain Structure

The Codomain Factor Spaces

At the base of the codomain structure lie the two codomain factor spaces W1 and W2, the individual target spaces of the maps f and g forming the factor map pair, each retaining its own identity as an ordinary vector space independent of any tensoring operation.

The Codomain Tensor Product

Built from the codomain factor spaces, the codomain structure includes the tensor product

the single space into which the induced map f tensor g delivers its outputs, formed from W1 and W2 according to the ordinary tensor product construction.

Internal Organization of the Codomain Structure

Elementary Tensor Outputs as the Generating Layer

The codomain structure is organized around elementary tensors of the form f(v) tensor g(w), since these are the direct outputs of the elementary output rule and generate, together with linear combinations, the entire image of the tensor product of maps within the codomain structure.

General Elements as the Complete Layer

Beyond these elementary tensor outputs, the codomain structure includes every element of W1 tensor W2 reachable as a finite sum of elementary tensors, whether or not that element actually arises as an output of f tensor g, since the codomain structure as a whole is simply the full tensor product space, independent of which elements happen to lie in the image.

Codomain Structure in Coordinates

Basis Assembly

Once bases are fixed for W1 and W2, the codomain structure acquires a concrete coordinate description: a basis of the codomain tensor product is assembled directly from the elementary tensors of basis vectors of W1 and W2, and every element of the codomain structure is represented, relative to this basis, by a single coordinate vector indexed by pairs of basis indices.

Dimension of the Codomain Structure

For finite-dimensional codomain factor spaces of dimensions p and q, the codomain structure has total dimension p q, matching the number of rows in the Kronecker product matrix used to represent the induced tensor product map once a corresponding domain structure is also fixed.

Codomain Structure Under Composition and Image

Stability of the Codomain Structure Under Composition

When composing two tensor products of maps, the codomain structure of the composite is exactly the codomain structure of the second tensor product of maps in the chain, since composition only modifies which map produces the output landing in the codomain structure, not the codomain structure itself.

Relationship Between the Codomain Structure and the Image

The image of a tensor product of maps is generally a proper subspace of the full codomain structure rather than the entire space, with the rank formula establishing exactly how much of the codomain structure is filled: the image has dimension equal to the product of the ranks of f and g, leaving the remainder of the codomain structure unreached whenever either f or g fails to be surjective on its own factor.