2.22 Tensor Linear Map Structure

Tensor Linear Map Structure is the collection of ways that linear maps between vector spaces interact with tensor constructions, comprising the identification of linear maps themselves as elements of a tensor product space, the natural operations of composing, adding, and scaling such maps inside that tensor description, and the induced action of a linear map on the various tensor spaces, tensor powers, symmetric powers, and exterior powers, built from its domain and codomain. This structure is what connects the abstract algebra of tensors to the concrete algebra of linear maps and matrices, and it is the reason tensor methods apply directly to problems phrased in terms of linear transformations.

Linear Maps as Tensors

The Identification V* ⊗ W ≅ Hom(V, W)

For finite-dimensional vector spaces V and W, there is a canonical isomorphism between the tensor product V* ⊗ W and the space Hom(V, W) of linear maps from V to W. A simple tensor ω ⊗ w, for ω ∈ V* and w ∈ W, corresponds to the rank-one linear map:

and this correspondence extends linearly to a bijection between all of V* ⊗ W and all of Hom(V, W), since both spaces have dimension dim(V) × dim(W) and the simple tensors e^i ⊗ f_j, built from a basis {e_i} of V and its dual {e^i}, together with a basis {f_j} of W, map to a basis of Hom(V, W) consisting of the elementary matrix maps.

Matrices as Coordinates of This Tensor

Once bases are fixed for V and W, the tensor Σ a^j_i e^i ⊗ f_j corresponds exactly to the linear map with matrix entries a^j_i, so that ordinary matrix notation is nothing more than the coordinate expression of an element of V* ⊗ W relative to the chosen bases. This is why a linear map is described by a (1, 1)-tensor: one lower index for the covector slot reading the input, one upper index for the vector slot producing the output.

Composition, Sum, and Scalar Multiplication

Composition as Contraction

If φ ∈ Hom(U, V) and ψ ∈ Hom(V, W) correspond to tensors α ∈ U* ⊗ V and β ∈ V* ⊗ W, the composite ψ ∘ φ ∈ Hom(U, W) corresponds to the tensor obtained by tensoring α and β and then contracting the upper index of α against the lower index of β:

which is precisely the ordinary rule for matrix multiplication; matrix multiplication is a special case of tensor contraction applied to the (1, 1)-tensor description of linear maps.

Sum and Scalar Multiplication

Because V* ⊗ W is a vector space, the sum of two linear maps and the scalar multiple of a linear map correspond directly to the sum and scalar multiple of the corresponding tensors, matching the ordinary pointwise definitions (φ + ψ)(v) = φ(v) + ψ(v) and (cφ)(v) = cφ(v) used for linear maps. Every vector-space operation on Hom(V, W) is therefore already accounted for by the tensor-space structure of V* ⊗ W, with no separate definition required.

Higher-Rank Linear Map Tensors

Multilinear Maps as Higher Tensors

A bilinear map B : V × V → F corresponds to an element of V* ⊗ V*, a (0, 2)-tensor, since it is linear in each argument separately and is completely determined by its values B(e_i, e_j) on pairs of basis vectors. More generally, a k-linear map V × ⋯ × V → F corresponds to an element of (V*)^{⊗k}, and a k-linear map into W rather than into the scalar field corresponds to an element of (V*)^{⊗k} ⊗ W, extending the Hom(V, W) ≅ V* ⊗ W identification to the fully multilinear setting.

Rank of the Underlying Linear Map

When V = W, an element of V* ⊗ V ≅ Hom(V, V) is an endomorphism, and its expression as a sum of simple tensors ω_1 ⊗ v_1 + ⋯ + ω_r ⊗ v_r with the fewest terms possible has r equal to the ordinary linear-algebra rank of the endomorphism; the minimal number of simple tensors needed to write a (1, 1)-tensor is exactly the rank of the linear map it represents, giving tensor rank and matrix rank a shared meaning for this particular type of tensor.

Induced Maps on Higher Tensor Spaces

The General Recipe

Given φ ∈ Hom(V, W), the induced map φ^{⊗k} ∈ Hom(T^k(V), T^k(W)) sends a simple tensor v_1 ⊗ ⋯ ⊗ v_k to φ(v_1) ⊗ ⋯ ⊗ φ(v_k), and this recipe restricts consistently to Sym^k(V) → Sym^k(W) and Λ^k(V) → Λ^k(W). When φ is not itself an isomorphism, these induced maps remain well defined linear maps, though they need not be injective or surjective even when φ is injective, in the case of Λ^k, or fails to be injective, in which case the induced map on any tensor power also fails to be injective.

Matrix Description on Tensor Powers

Relative to bases, if φ has matrix a^j_i, the induced map φ^{⊗k} has matrix entries given by the product a^{j_1}_{i_1} ⋯ a^{j_k}_{i_k} indexed by the multi-indices (i_1, ..., i_k) and (j_1, ..., j_k), so the matrix of the induced map on a tensor power is built entirely from products of entries of the original matrix, with no additional data required.

Why Linear Map Structure Is Part of Tensor Algebra

A Unifying Perspective

Treating linear maps, bilinear forms, and higher multilinear maps all as elements of appropriate tensor product spaces removes the need for a separate theory of matrices, composition, and multilinear algebra: matrix multiplication is contraction, the transpose of a matrix corresponds to swapping index roles, and the rank of a map is the tensor rank of its (1, 1)-tensor representative. Tensor linear map structure is the set of facts that makes this unification precise and computationally usable.