12.13.2 Tensor Projection Target Component

Tensor Projection Target Component is the specific piece of the target space that a tensor projection operation isolates and returns, namely the element π(t) ∈ T′ produced by applying the projection π : T → T′ to a particular tensor t of the source space. Where the target space T′ describes the ambient space of all possible outputs of a given projection, the target component is the individual output associated with one specific input tensor, together with whatever internal structure — index labeling, factor identity, or basis coordinates — identifies exactly which piece of t it represents.

Identity of the Target Component

As the Image of a Single Input

For a fixed projection π and a fixed input tensor t, the target component is simply π(t), an element of T′ determined uniquely by t and by the choice of π. Changing either the input tensor or the projection itself changes which target component results, so the target component is always relative to both of these choices, not an intrinsic property of T′ alone.

Labeling by Factor or Index

When the projection extracts one factor of a tensor product, π : V₁ ⊗ V₂ → V₁, the target component of v₁ ⊗ v₂ is labeled as "the V₁-component" of that simple tensor, namely v₁ itself (up to any scalar produced by evaluating against V₂). When the projection instead contracts a fixed index slot of a higher-rank tensor, the target component is labeled by which index was contracted and which indices remain, since a single tensor typically admits several different target components depending on which projection is applied.

Relationship to the Complementary Component

Decomposition into Component and Remainder

When π is idempotent on a single space T, every tensor t splits into its target component π(t) and a complementary remainder t − π(t) lying in the kernel of π. The target component is precisely the summand retained by the projection, while the remainder is the summand discarded; together they reconstruct the original tensor exactly, with no overlap and no loss.

The Target Component Is Fixed by Further Projection

Because π ∘ π = π, applying π to the target component itself reproduces the same target component: π(π(t)) = π(t). This confirms that the target component, once extracted, is already "purely" what the projection selects — no further residue of the discarded remainder is hidden inside it.

Diagram of a Tensor Split into Target Component and Remainder

Target Components Across Multiple Projections

Simultaneous Extraction of Several Components

If a family of projections π₁, ..., πₙ satisfies πᵢπⱼ = 0 for i ≠ j and Σᵢ πᵢ = id_T, every tensor t decomposes into n simultaneous target components π₁(t), ..., πₙ(t), one for each projection in the family, summing exactly back to t. Each target component then represents t's contribution to one summand of a full direct-sum decomposition of T.

Coordinates as a Special Case

When the projections πᵢ extract the coefficient of each basis vector eᵢ in a fixed basis of T, the resulting target components are exactly the coordinates of t relative to that basis. Ordinary tensor coordinates are therefore a special case of target components, arising from the particular family of coordinate-extraction projections associated with a chosen basis.

Sensitivity to the Choice of Projection

Different Projections Yield Different Target Components for the Same Tensor

The same tensor t generally has a different target component for each distinct projection applied to it; there is no universal "the component of t" independent of specifying which projection produced it. Reporting a target component without specifying the projection that generated it leaves the description incomplete, since the same numeric or symbolic value could arise from unrelated projections applied to unrelated tensors.

Basis-Dependence Inherited from the Projection

If the projection itself was only definable relative to a chosen basis or a chosen auxiliary functional, the resulting target component inherits that same dependence. A target component computed under one basis choice need not equal, or even correspond term-by-term to, the target component computed for the same abstract tensor under a different, equally valid basis choice for the same projection family.