5.19.4 Tensor Product Vector Space Basis Construction

Constructing a basis for the tensor product space by combining bases of the original vector spaces through bilinear mappings and direct sum structures.

Tensor Product Vector Space Basis Construction is the explicit procedure by which a basis of the tensor product V ⊗ W of two vector spaces is built from a chosen basis of V and a chosen basis of W, together with the verification that the resulting family of simple tensors is linearly independent and spanning regardless of which bases of V and W were originally selected. This construction is the standard concrete route to the tensor product, complementing the abstract universal-property definition with an explicit, computable object that can be used for coordinates, matrix representations, and dimension counting.

The Construction Procedure

Choosing Bases of the Factors

Let V and W be vector spaces over a field F, with bases {eᵢ}_{i∈I} for V and {fⱼ}_{j∈J} for W, where I and J are arbitrary index sets (finite or infinite). No compatibility between the two bases is required; they are chosen independently.

Forming the Candidate Family

The candidate basis of V ⊗ W is the family of simple tensors

{e_{i} \otimes f_{j} : i \in I, j \in J}

indexed by the Cartesian product I × J. Every element of this family is a simple (decomposable) tensor, and the construction's content is the claim that this particular family constitutes a basis of the whole tensor product, not merely a spanning set of simple tensors.

Verifying the Spanning Property

Reducing an Arbitrary Element to a Finite Sum

An arbitrary element of V ⊗ W is, by definition of the tensor product, a finite sum of simple tensors Σ vₖ ⊗ wₖ. Expanding each vₖ in the basis {eᵢ} and each wₖ in the basis {fⱼ} and using bilinearity of ⊗ to distribute the sums produces

\sum_{k} v_{k} \otimes w_{k} = \sum_{i, j} c_{i j} (e_{i} \otimes f_{j})

for scalars c_{ij} that are nonzero for only finitely many pairs (i,j). This shows every element of V ⊗ W lies in the span of {eᵢ ⊗ fⱼ}.

Bilinearity as the Mechanism

The only property of ⊗ used in this reduction is bilinearity: (Σaᵢeᵢ) ⊗ (Σbⱼfⱼ) = Σᵢⱼ aᵢbⱼ (eᵢ ⊗ fⱼ). No further structural fact about the tensor product is needed to establish spanning, which is why spanning is the easier of the two properties to verify.

Verifying Linear Independence

The Coordinate Functional Argument

Linear independence requires more than bilinearity; it invokes the universal property. Fix a pair of indices (k, l) ∈ I × J. Define a bilinear map β_{kl} : V × W → F on pairs (v, w) = (Σ aᵢeᵢ, Σ bⱼfⱼ) by

β_{k l} (v, w) = a_{k} b_{l}

By the universal property of the tensor product, β_{kl} factors uniquely through a linear functional φ_{kl} : V ⊗ W → F satisfying φ_{kl}(eᵢ ⊗ fⱼ) = δ_{ik} δ_{jl}, the Kronecker delta.

Isolating a Single Coefficient

Suppose a finite linear combination vanishes, Σᵢⱼ c_{ij}(eᵢ ⊗ fⱼ) = 0. Applying φ_{kl} to both sides and using linearity gives

0 = φ_{k l} (\sum_{i, j} c_{i j} (e_{i} \otimes f_{j})) = c_{k l}

Since (k, l) was arbitrary, every coefficient c_{ij} must vanish, establishing linear independence of the whole family.

Why the Universal Property Is Indispensable Here

Spanning followed from bilinearity alone, but independence required manufacturing, for each pair of indices, a linear functional that "detects" that coordinate and ignores all others. Constructing such detecting functionals is precisely what the universal property guarantees is possible, which is why independence — unlike spanning — cannot be proved from bilinearity in isolation.

Diagram of the Basis Correspondence

Independence from the Choice of Bases

Different Bases Yield Isomorphic Basis Families

If {e′ᵢ} and {f′ⱼ} are alternative bases of V and W, the family {e′ᵢ ⊗ f′ⱼ} is also a basis of V ⊗ W, obtained from {eᵢ ⊗ fⱼ} by the corresponding basis change. The construction therefore does not privilege any particular basis of V ⊗ W; it produces a basis for every choice of input bases, all related by the induced change-of-basis transformation.

Consequence for Well-Definedness of the Tensor Product

Because the spanning and independence arguments above never used any special property of the chosen bases beyond their being bases, the construction confirms that V ⊗ W is well-defined as an abstract object of dimension dim(V) · dim(W) independent of which bases were used to exhibit it, reconciling the concrete basis construction with the basis-free universal-property definition.

Extensions and Practical Uses

Extension to Several Factors

The same construction extends directly to a tensor product of finitely many spaces V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ: choosing a basis for each factor, the family of all simple tensors e^{(1)}_{i₁} ⊗ e^{(2)}_{i₂} ⊗ ... ⊗ e^{(n)}_{iₙ} indexed over the product of index sets forms a basis of the multi-factor tensor product, by an identical spanning-and-independence argument applied iteratively.

Coordinates and Matrix Representation

Once the basis {eᵢ ⊗ fⱮ} is fixed, every element of V ⊗ W acquires coordinates c_{ij}, which for finite-dimensional V and W can be arranged into an m × n matrix; this is the basis-dependent bridge between abstract tensors and the concrete matrices used throughout linear algebra and its applications.

The Infinite-Dimensional Case

When I or J is infinite, the same spanning and independence arguments go through verbatim, since only finite linear combinations are ever involved; the resulting basis {eᵢ ⊗ fⱼ} is a genuine Hamel basis of V ⊗ W indexed by I × J, confirming the construction's validity outside the finite-dimensional setting as well.