12.14.3 Tensor Inclusion Embedding Map Role

Tensor Inclusion Embedding Map Role is the requirement that a tensor inclusion operation act as a genuine embedding, meaning it must be simultaneously linear and injective, so that the source subspace is not merely mapped into the target space but is realized there as a faithful, structurally identical copy of itself. This role is what separates an inclusion from an arbitrary linear map into a larger space: any linear map can send U into T, but only an embedding preserves enough information to let U be treated, without qualification, as literally sitting inside T.

Components of the Embedding Map Role

Linearity

Linearity ensures that sums and scalar multiples formed in U correspond exactly to sums and scalar multiples formed in T after embedding. Without this, ι(U) could fail to be a subspace of T at all, undermining any claim that U has been faithfully placed inside T.

Injectivity

For a linear map, injectivity is equivalent to having trivial kernel: the only element of U sent to the zero tensor of T is 0 itself. This is the condition that specifically earns the map the title "embedding" rather than merely "linear map," since a linear map with nontrivial kernel collapses distinct elements of U together and cannot faithfully represent U's internal structure inside T.

The Two Conditions Are Independent

A map can be linear without being injective, and — outside the linear setting — injective without being linear; the embedding map role requires both simultaneously. Neither condition alone suffices: a linear map with nontrivial kernel is not an embedding, and a merely injective but nonlinear map is not usable in the tensor-algebraic constructions that inclusion is meant to support.

Consequences of Satisfying the Embedding Map Role

Isomorphism onto the Image

Once ι is both linear and injective, it restricts to a linear isomorphism from U onto its image ι(U) ⊆ T. Every property of U expressible in terms of its own linear structure — dimension, basis cardinality, solvability of linear systems — transfers without alteration to ι(U), and conversely any such property observed in ι(U) can be pulled back faithfully to U.

Well-Defined Inverse on the Image

Because ι is injective, the map ι⁻¹ : ι(U) → U is well-defined on the image, even though ι itself need not be invertible as a map into all of T. This partial inverse is what allows computations performed inside T, once restricted to the subspace ι(U), to be translated back into statements about U without ambiguity.

Preservation of Linear Independence

If {fₖ} is a linearly independent set in U, the embedding map role guarantees {ι(fₖ)} is linearly independent in T. A failure of injectivity would allow a nontrivial linear combination of the fₖ to map to zero, destroying this independence; the embedding role is exactly what rules this out.

Diagram of the Embedding Map Role

Verifying the Embedding Map Role in Practice

Checking Linearity via a Basis

Linearity of ι is verified by confirming ι(afₖ + bfⱼ) = aι(fₖ) + bι(fⱼ) on a spanning set {fₖ} of U, together with the assumption that ι is defined by extending its values on this spanning set linearly. This reduces an infinite family of required identities to a finite check whenever U is finite-dimensional.

Checking Injectivity via the Kernel Computation

Injectivity is verified by solving ι(u) = 0 for u ∈ U and confirming the only solution is u = 0. In coordinates, if ι is represented by a matrix M relative to bases of U and T, this amounts to confirming M has full column rank, equal to dim(U).

Failure Diagnosis

If the kernel computation produces a nonzero solution u₀, the candidate map fails the embedding map role, and u₀ together with −u₀, or any two distinct elements differing by u₀, furnish an explicit witness that distinct elements of U are being sent to the same element of T, disqualifying the map as a tensor inclusion operation.

Relation to Other Tensor Operation Roles

Contrast with the Projection's Linear Map Role

The linear map role required of a tensor projection demands only linearity, since a projection is expected to be many-to-one on its kernel by design. The embedding map role required of an inclusion additionally demands injectivity, reflecting the opposite intention: an inclusion must lose no information, while a projection is explicitly permitted, and often intended, to discard some.

Compatibility with Paired Retraction

The embedding map role is precisely what is needed for the retraction identity π ∘ ι = id_U to be achievable by some projection π: without injectivity, no linear map π could reconstruct u from ι(u) for every u, since the value ι(u) would not uniquely determine u in the first place.