14.3.4 Tensor Map Product Linearity Preservation

Tensor Map Product Linearity Preservation ensures that tensor products maintain linearity across mappings, preserving structural integrity in algebraic transformations.

Tensor Map Product Linearity Preservation is the property that the induced map f tensor g obtained from a tensor product of maps is itself a linear map on the tensor product of the domains, meaning it respects addition and scalar multiplication of the domain tensor space rather than merely acting predictably on elementary tensors.

Statement of the Property

Additivity on the Tensor Product

Linearity preservation requires that for all elements t and s of V1 tensor V2,

(f \otimes g) (t + s) = (f \otimes g) (t) + (f \otimes g) (s),

holding for arbitrary elements of the tensor product, not merely for elementary tensors, which is the substantive content of the property since the defining formula is only given directly on elementary tensors.

Homogeneity on the Tensor Product

Linearity preservation likewise requires

(f \otimes g) (λ t) = λ (f \otimes g) (t),

for every scalar lambda and every element t of the tensor product, again for general elements rather than only elementary ones.

Source of the Preservation

Direct Consequence of the Universal Property

Linearity preservation is not an additional fact requiring separate verification once f tensor g is constructed through the universal property of the tensor product: the universal property produces f tensor g specifically as a linear map, so linearity on the whole tensor space is built into the existence statement itself rather than needing to be checked afterward on general elements.

Contrast with the Elementary Formula Alone

If one attempted to define f tensor g only by declaring its values on elementary tensors and extending "by linearity" without appeal to the universal property, linearity preservation would need to be verified directly, by checking that the resulting rule assigns consistent values regardless of how a given element is decomposed into a sum of elementary tensors. The universal property bypasses this verification by guaranteeing consistency automatically.

Preservation Under Operations

Preservation Under Composition

If f tensor g and f prime tensor g prime are each linearity-preserving maps on their respective tensor products, then their composite is linearity-preserving as well, since a composite of linear maps is linear, and the composition identity

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g)

confirms that this composite is again a tensor product of maps, and hence itself linearity-preserving by the same universal-property argument.

Preservation Under Sums and Scalar Multiples of the Factor Maps

If f1 and f2 are both linear maps sharing the same domain and codomain as f, then f1 plus f2 tensor g is again linearity-preserving, since it too arises from the universal property applied to a suitable bilinear map, and the identity

(f_{1} + f_{2}) \otimes g = (f_{1} \otimes g) + (f_{2} \otimes g)

expresses this new linearity-preserving map as a sum of two linearity-preserving maps, itself linear as a sum of linear maps.

Consequences for Matrix Representation

Reflection in the Kronecker Product

Linearity preservation is what justifies representing f tensor g by a single matrix, the Kronecker product of the matrices of f and g, with respect to a chosen basis: only a linear map admits such a matrix representation acting correctly on every element of the domain, not merely on the basis elementary tensors used to build the matrix in the first place.

Necessity for Rank and Spectral Formulas

The rank, trace, determinant, and eigenvalue formulas established for f tensor g all presuppose linearity preservation, since these are invariants defined for linear maps; without linearity holding on the entire tensor product, rather than only on elementary tensors, such formulas would not be meaningful statements about f tensor g as a map.