5.4.4 Tensor Product Space Basis Dependence

Tensor Product Space Basis Dependence is the classification of which facts about V ⊗ W require a choice of basis of V and W to state, and which hold intrinsically, independent of any such choice, marking a boundary that runs through the space's structure between coordinate-dependent computation and basis-free definition.

What Is Basis-Independent

The Carrier Set, Addition, and Scalar Action

The cosets making up the carrier set of V ⊗ W, the addition of two such cosets, and the scalar action of the field on a coset are all defined directly from the quotient F(V × W)/R, with no reference anywhere in their definitions to a basis of V or W; two different choices of basis for V and W do not change a single element of V ⊗ W, nor how any two elements add, nor how a scalar acts on any element.

The Universal Property and Dimension as an Abstract Count

The universal factorization property of V ⊗ W is stated purely in terms of maps and holds regardless of basis; likewise, the equality dim(V ⊗ W) = dim(V) · dim(W) is basis-independent as a numerical fact, even though the standard proof of it proceeds by exhibiting a specific induced basis.

Decomposability of a Given Element

Whether a specific element t of V ⊗ W is decomposable — expressible as a single v ⊗ w — is a basis-independent property of t; it does not become decomposable in one basis and non-decomposable in another, since decomposability is a statement about the existence of vectors v, w with t = v ⊗ w, unaffected by how coordinates happen to be assigned.

What Is Basis-Dependent

The Specific Induced Basis {e_i ⊗ f_j}

Once bases {e_i} of V and {f_j} of W are chosen, the induced set {e_i ⊗ f_j} is a specific basis of V ⊗ W; choosing different bases of V or W produces a different induced basis of V ⊗ W, related to the first by the same change-of-basis transformation applied to each factor, rather than an arbitrary unrelated basis of the tensor product.

Coordinate Arrays

The coefficients c_{ij} expressing a given element t as ∑ c_{ij}(e_i ⊗ f_j) depend entirely on the chosen bases; the same element t has a different coordinate array relative to a different pair of bases, transforming according to the tensor product of the two individual change-of-basis matrices.

Matrix Representation of Tensor Rank Computations

While tensor rank itself — the minimal number of decomposable summands — is a basis-independent invariant of an element, the practical method of computing it for two-factor tensor products by identifying V ⊗ W with matrices and taking matrix rank is basis-dependent in its presentation, though the resulting numerical rank value it computes does not change under a change of basis, since a change of basis corresponds to left and right multiplication by invertible matrices, which does not alter matrix rank.

Why the Distinction Matters

Separating What Must Be Proved Once from What Must Be Recomputed Each Time

Facts established without reference to a basis, such as the universal property or the abstract dimension formula, need be proved only once and apply uniformly; facts stated in coordinates, such as a specific numerical coordinate array for an element, must be recomputed whenever the basis changes. Basis dependence classification is what tells a reader, for any given fact about V ⊗ W, whether a change of basis on V or W requires redoing the associated computation or leaves the stated fact untouched.