4.24.1 Tensor Linear Map Boundary

Tensor Linear Map Boundary is the delineation of the single-argument case of the multilinear map boundary, fixing exactly which maps qualify as linear maps for the purpose of tensor identification: functions of one vector argument that preserve vector addition and scalar multiplication, taking values in the base field or a designated target vector space. As the k = 1 instance of multilinearity, the linear map boundary is the narrowest and most elementary of the arity-indexed boundaries, and it anchors the tensor correspondence between linear maps and type (1, 1) or (0, 1) tensors.

The Defining Condition

Additivity and Homogeneity

A map

or, for vector-valued linear maps,

lies inside the boundary when it satisfies

and

for all vectors u, v in V and all scalars c in F. Because there is only one argument, additivity and homogeneity are the entirety of the boundary condition; there is no cross-argument requirement to state, which is what makes the linear case the base of the multilinear hierarchy rather than merely one instance among equals.

Consequence: Determination by Basis Images

A direct consequence internal to the boundary is that a linear map is completely determined once its values on a basis of V are fixed, since every other vector decomposes uniquely as a finite linear combination of basis vectors and T is forced to respect that combination. This finite determination is what allows a linear map on a finite-dimensional V to be recorded by finitely many field elements or vectors, matching it with a specific tensor of type (1, 1).

What Falls Outside the Boundary

Affine Maps

A map of the form T(v) = A(v) + b with b a fixed nonzero vector fails homogeneity, since T(0) is b rather than 0; such affine maps are excluded from the linear boundary even though they are built from a genuinely linear part A. The exclusion is strict: no adjustment of the domain or codomain brings an affine map with b ≠ 0 inside the boundary, because the failure is intrinsic to the map itself.

Maps Depending on Norm or Absolute Value

Any map whose definition involves a norm, absolute value, or other non-additive function of the input, such as T(v) = |v| for real scalars, fails additivity in general and is excluded regardless of whether the map happens to preserve scalar multiplication for positive scalars.

Relation to Adjacent Boundaries

Special Case of the Multilinear Boundary

Tensor Linear Map Boundary sits inside Tensor Multilinear Map Boundary as the k = 1 restriction: every condition imposed there — separate linearity in each argument, a field or designated vector space codomain — collapses to the ordinary linear map condition when there is only one argument to be linear in. No new requirement is introduced at this level; the boundary is a specialization, not an extension.

Building Block for Bilinear and Higher Boundaries

A bilinear map can be viewed, argument by argument, as a family of linear maps: fixing the first argument of a bilinear map produces a linear map in the second, and vice versa. The linear map boundary therefore functions as the elementary unit from which the bilinear and general multilinear boundaries are assembled, one fixed-argument slice at a time.