2.22.1 Tensor Linear Map Domain Space

Tensor Linear Map Domain Space is the vector space V that supplies the input side of a linear map φ : V → W, considered specifically in its role inside the tensor identification Hom(V, W) ≅ V* ⊗ W, where the domain space contributes its dual V* rather than V itself to the tensor product representing the map. Understanding the domain space in this role clarifies why the input side of a linear map always appears as a covariant, lower-index factor in the tensor description, and how properties of the domain, such as its dimension and choice of basis, govern the corresponding factor of the tensor.

Why the Domain Contributes Its Dual

Reading Off a Vector Requires a Covector

A linear map φ acts on an input vector v ∈ V by combining v with data drawn from V*, since the way a rank-one map ω ⊗ w acts is (ω ⊗ w)(v) = ω(v) w, which first reads v through the linear functional ω and then scales the output vector w by the resulting number. This is why the domain space enters the tensor description of Hom(V, W) through V* rather than through V: extracting a definite number from an input vector, so that it can multiply an output vector, is exactly what an element of the dual space does.

Contrast With the Codomain

The codomain W enters the same tensor description directly, without dualizing, because the codomain simply supplies the vector-valued output of the map. The domain and codomain therefore play asymmetric roles in the (1, 1)-tensor representing a linear map: the domain contributes a covariant, lower-index slot, and the codomain contributes a contravariant, upper-index slot, matching the general convention that lower indices consume inputs and upper indices produce outputs.

Dimension and Basis of the Domain Space

Dimension Contribution

If dim(V) = n, then dim(V*) = n as well, so the domain space contributes a factor of n to the overall dimension dim(V* ⊗ W) = dim(V) × dim(W) of the space of linear maps; the domain and the codomain contribute their dimensions symmetrically to this product even though they enter the tensor product asymmetrically, through V* and W respectively.

Dual Basis From a Domain Basis

Fixing a basis {e_i} for the domain space V determines a corresponding dual basis {e^i} for V*, characterized by:

and it is this dual basis of the domain, tensored against a basis {f_j} of the codomain, that produces the standard basis {e^i ⊗ f_j} of V* ⊗ W used to write down matrix entries; the lower index i of a matrix entry a^j_i is indexed by the domain's dual basis, and the upper index j by the codomain's basis.

Effect of a Change of Domain Basis

Transformation of the Covariant Slot

If the domain basis is changed from {e_i} to {e'_i}via a transition matrix A, the dual basis transforms with A directly, since dual bases are covariant, and consequently the domain-associated lower index of the tensor a^j_i describing the linear map transforms according to the same covariant rule used generally for lower indices in coordinate transfer:

Independence From the Codomain Basis

Because the domain and codomain contribute independent tensor factors, a change of basis in the domain space affects only the lower index of the map's matrix, leaving the upper, codomain-associated index untouched, and conversely a change of basis in the codomain affects only the upper index; the two factors of V* ⊗ W transform independently under separate changes of basis in V and W.

The Domain Space in Higher Multilinear Constructions

Multiple Domain Factors

For a k-linear map V × ⋯ × V → W, corresponding to an element of (V*)^{⊗k} ⊗ W, the domain space contributes one dual factor for each of its k argument slots, so a bilinear map has two lower indices drawn from the domain and one upper index from the codomain. Each argument slot of the domain is tracked by its own separate copy of V* in the tensor product, allowing the map to depend independently on each input.

Role in Contraction and Composition

When composing linear maps, the codomain of one map must match the domain of the next, and this matching is precisely what permits the contraction described in tensor linear map structure: the upper index contributed by the first map's codomain is contracted against the lower index contributed by the second map's domain, since contraction requires pairing an upper index with a lower index built from dual copies of the same underlying space.