5.2.3 Tensor Product Basis Area

The tensor product basis area explains how tensor products generate bases for multilinear spaces, essential in algebra and physics.

Tensor Product Basis Area is the detailed treatment of how bases of V and W induce a basis of V ⊗ W, covering the proof that the induced set spans the space, the proof that it is linearly independent, and the resulting dimension formula and coordinate expansion for an arbitrary element.

Constructing the Induced Set

The Candidate Basis

Let {e_1, ..., e_m} be a basis of V and {f_1, ..., f_n} be a basis of W. The candidate basis for V ⊗ W is the set of mn decomposable elements

\{e_{i} \otimes f_{j}\} 1 \leq i \leq m, 1 \leq j \leq n

one for every pairing of a basis vector of V with a basis vector of W.

Proving the Set Spans

Every Decomposable Element Expands

For arbitrary v = ∑_i a_i e_i in V and w = ∑_j b_j f_j in W, bilinearity of ⊗ gives

v \otimes w = \sum_{i} \sum_{j} a_{i} b_{j} (e_{i} \otimes f_{j})

so every decomposable element is a linear combination of the induced set.

Every General Element Expands

Since every element of V ⊗ W is a finite sum of decomposable elements, and each such term expands as above, every element of V ⊗ W is a linear combination of {e_i ⊗ f_j}; the induced set therefore spans V ⊗ W.

Proving Linear Independence

Using the Universal Property to Detect Coefficients

Suppose ∑_{i,j} c_{ij} (e_i ⊗ f_j) = 0. Fix indices p, q, and let e_p* and f_q* be the coordinate functionals dual to e_p and f_q with respect to the chosen bases. The bilinear map (v, w) ↦ e_p*(v) f_q*(w) induces, by the universal property, a linear functional φ_{pq} on V ⊗ W with φ_{pq}(e_i ⊗ f_j) = δ_{ip} δ_{jq}.

Applying the Functional

Applying φ_{pq} to the assumed relation gives

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

for every choice of p, q, so every coefficient vanishes and the induced set is linearly independent. This is where the universal property, not just the spanning argument, is essential: it supplies the coordinate-extraction functionals used to isolate each coefficient.

Consequences

Dimension Formula

Because {e_i ⊗ f_j} is both spanning and linearly independent, it is a basis of V ⊗ W, so dim(V ⊗ W) = dim(V) · dim(W) = mn, in contrast to the direct sum V ⊕ W, whose dimension is m + n.

Coordinate Expansion Is Unique

Every element of V ⊗ W is a unique linear combination ∑_{i,j} c_{ij} (e_i ⊗ f_j); the coefficients c_{ij} form an m × n array, which is the coordinate representation used whenever a specific basis-dependent description of a tensor product element is required.