6.15.3 Tensor Two Zero Basis Product Relation

The Tensor Two Zero Basis Product Relation explores how zero basis elements interact in tensor algebra, foundational for understanding tensor product structures.

Tensor Two Zero Basis Product Relation is the identity expressing an arbitrary type (2,0) tensor as a linear combination of elementary basis products e_i ⊗ e_j, formed by taking every possible ordered pair of basis vectors from the underlying vector space, and showing that these n² products together constitute a complete basis for the tensor product space V ⊗ V. This relation is what converts the abstract construction of the tensor product into a concrete, computable object: once the basis product relation is established, every type (2,0) tensor reduces to a finite list of coefficients, namely its components, multiplying a fixed and universal set of building blocks.

Constructing the Basis Products

From a Basis of V to a Basis of V Tensor V

Given a basis {e_1, ..., e_n} of the vector space V, the basis product relation asserts that the collection of all ordered pairs:

e_{i} \otimes e_{j}

for i and j each ranging from 1 to n, forms a basis for V ⊗ V. Since there are n choices for i and n choices for j, this produces exactly n² basis elements, matching the dimension of the tensor product space, dim(V ⊗ V) = dim(V) · dim(V) = n².

Expansion of an Arbitrary Tensor

Any type (2,0) tensor T can be written uniquely as a linear combination of these basis products, with the components T^{ij} serving as the coefficients:

T = T^{i j} e_{i} \otimes e_{j}

with the repeated indices i and j summed according to the summation convention. This relation is the precise sense in which the components T^{ij} "are" the tensor once a basis has been fixed: they are the unique coordinates of T relative to the basis product family {e_i ⊗ e_j}.

Why the Basis Products Are Linearly Independent and Spanning

Linear Independence

The basis products e_i ⊗ e_j are linearly independent because the tensor product construction is designed precisely so that a sum Σ c_{ij} (e_i ⊗ e_j) vanishes only when every coefficient c_{ij} is zero; this follows from the universal property of the tensor product, which guarantees that bilinear maps out of V × V correspond exactly to linear maps out of V ⊗ V, and applying this correspondence to the coordinate functionals of V shows that no nontrivial linear relation can hold among the products e_i ⊗ e_j.

Spanning

Every elementary tensor v ⊗ w can be expanded using the bilinearity of the tensor product and the expansions v = v^i e_i and w = w^j e_j:

v \otimes w = v^{i} w^{j} (e_{i} \otimes e_{j})

and since general elements of V ⊗ V are finite sums of elementary tensors, the basis products span the entire space.

Behavior of the Relation Under a Change of Basis

New Basis Products from New Basis Vectors

If the basis of V is changed from {e_i} to {e'_i} via a transition matrix A, so that e'_i = A^k_i e_k, then the new basis products of V ⊗ V are obtained by substituting directly:

e_{' i} \otimes e_{' j} = A_{i}^{k} A_{j}^{l} (e_{k} \otimes e_{l})

Consistency with Component Transformation

Because the tensor T itself does not change when the basis changes, only its expression, the coefficients in the new basis product relation must transform inversely to the way the basis products themselves transform, which is exactly the origin of the contravariant transformation law T'^{ij} = B^i_k B^j_l T^{kl} for the components. The basis product relation and the component transformation law are therefore two faces of the same underlying fact: T is invariant, and any change in how it is expressed as a combination of basis products must be compensated by an inverse change in the coefficients.

Diagram of the Product Basis Construction

Consequences of the Basis Product Relation

Dimension Counting for Symmetric and Antisymmetric Subspaces

Because the full basis product family has n² elements split into n "diagonal" products e_i ⊗ e_i and n(n-1) "off-diagonal" products e_i ⊗ e_j with i ≠ j paired up into n(n-1)/2 symmetric combinations and the same number of antisymmetric combinations, the basis product relation directly yields the dimension formulas n(n+1)/2 for the symmetric subspace and n(n-1)/2 for the antisymmetric subspace of V ⊗ V.

Foundation for Higher-Order Products

The same construction extends recursively: a basis for V ⊗ V ⊗ V is obtained by forming triple products e_i ⊗ e_j ⊗ e_k of basis vectors, and in general the basis product relation for any tensor power of V is built by taking all ordered tuples of basis vectors, making the type (2,0) case the smallest nontrivial instance of a pattern that underlies the construction of tensors of arbitrary order.