2.21.1 Tensor Isomorphism Linear Map

Tensor Isomorphism Linear Map is the specific linear transformation φ : V → W that witnesses an isomorphism between two vector spaces, examined here as a concrete map with verifiable criteria, algebraic properties, and an automatically linear inverse, rather than as the abstract fact of isomorphism between spaces. Where the vector space isomorphism structure classifies isomorphisms as canonical or non-canonical and studies their induced action across tensor constructions, the isomorphism linear map narrows the focus to a single such map: what conditions a linear map must satisfy to qualify as an isomorphism, how those conditions are checked in coordinates, and how the map itself, not merely its existence, is built and composed.

Criteria for a Linear Map to Be an Isomorphism

Bijectivity as the Defining Condition

A linear map φ : V → W is an isomorphism precisely when it is bijective, meaning both injective and surjective. Because φ is linear, these two conditions can each be restated in terms of simpler linear-algebraic data rather than checked directly as set-theoretic injectivity and surjectivity.

Injectivity via the Kernel

φ is injective if and only if its kernel is trivial, ker(φ) = {0}. This follows because linearity makes injectivity equivalent to the statement that only the zero vector maps to zero: if φ(u) = φ(v), then φ(u - v) = 0 by linearity, so u = v exactly when the kernel contains no nonzero vector.

Surjectivity and the Rank-Nullity Theorem

For finite-dimensional V and W with dim(V) = dim(W), the rank-nullity theorem, dim(V) = dim(ker(φ)) + dim(im(φ)), shows that injectivity and surjectivity become equivalent: a trivial kernel forces im(φ) to have full dimension dim(W), hence im(φ) = W. This equivalence is why, in the equal-dimension setting relevant to tensor algebra, checking only injectivity, or only surjectivity, suffices to confirm that φ is a full isomorphism.

Matrix Criteria

Invertibility of the Representing Matrix

Once bases are fixed for V and W, the map φ is represented by a matrix M. φ is an isomorphism if and only if M is invertible, equivalently if and only if det(M) ≠ 0, equivalently if and only if the columns of M are linearly independent. Any of these equivalent matrix conditions gives a concrete, computable test for whether a given linear map is an isomorphism.

The Inverse Map's Matrix

When φ is an isomorphism, its inverse φ^{-1} : W → V is represented, in the same pair of bases, by the matrix inverse M^{-1}. This gives an explicit recipe for constructing the inverse map: compute the matrix of φ, invert it, and interpret the result as the matrix of φ^{-1}.

The Inverse of a Linear Isomorphism Is Automatically Linear

Linearity of φ⁻¹

If φ is a bijective linear map, its set-theoretic inverse φ^{-1} is automatically linear, not merely a bijective function. This follows directly from applying φ^{-1} to the linearity equations for φ: since φ(αu + βv) = αφ(u) + βφ(v) for all u, v and scalars α, β, substituting u = φ^{-1}(x) and v = φ^{-1}(y) and applying φ^{-1} to both sides shows φ^{-1}(αx + βy) = αφ^{-1}(x) + βφ^{-1}(y).

Why This Matters for Tensor Algebra

This automatic linearity is what allows φ^{-1} to be used freely wherever a linear map is required, for instance in defining the contragredient transformation of covectors, (φ^{-1})^*, without needing a separate argument that the inverse map respects vector space structure.

Explicit Construction via Basis Correspondence

Building an Isomorphism from Two Bases

Given a basis e_1, ..., e_n of V and a basis f_1, ..., f_n of W, an explicit isomorphism φ : V → W is constructed by declaring φ(e_i) = f_i and extending linearly, φ(v^i e_i) = v^i f_i. This construction always succeeds precisely because a linear map is uniquely determined by its values on a basis, and any assignment of basis vectors to basis vectors, extended linearly, automatically produces a bijection between the two spaces.

Non-Uniqueness of the Constructed Map

Different choices of basis correspondence generally produce different isomorphisms; there is no single "the" isomorphism between two abstractly isomorphic spaces of the same dimension, only a family of isomorphisms indexed by the choice of matching between the two bases. This non-uniqueness is exactly the phenomenon underlying the distinction, in the vector space isomorphism structure, between non-canonical isomorphisms such as V ≅ V* and the unique canonical isomorphism V ≅ V**.

Composition and Group Structure

Composition of Isomorphisms Is an Isomorphism

If φ : V → W and ψ : W → U are both isomorphisms, their composite ψ ∘ φ : V → U is again an isomorphism, with inverse φ^{-1} ∘ ψ^{-1}. This closure under composition, together with the existence of inverses and the identity map, is what makes the set of isomorphisms V → V, that is, GL(V), into a group.

Induced Maps on Associated Tensor Constructions

Given an isomorphism linear map φ : V → W, several further maps are induced automatically: the pullback φ^* : W* → V* on dual spaces, defined by φ^*(g) = g ∘ φ, and the tensor power map φ^{⊗k} : V^{⊗k} → W^{⊗k}, defined on simple tensors by φ^{⊗k}(v_1 ⊗ ... ⊗ v_k) = φ(v_1) ⊗ ... ⊗ φ(v_k) and extended linearly. Each of these induced maps is itself an isomorphism whenever φ is, which is the concrete, map-level mechanism by which an isomorphism of vector spaces propagates to an isomorphism of every tensor space built from them.