9.3.4 Tensor Basis Independence Requirement

Tensor Basis Independence Requirement is the condition, complementary to the span requirement, that no vector in a proposed basis collection can be written as a linear combination of the others, equivalently that the only linear combination of the collection summing to the zero vector is the one with every coefficient equal to zero — a condition whose role is to guarantee that the coefficients used to expand any given vector or tensor in the basis are unique, so that tensor components are well-defined numbers rather than one of several equally valid possibilities.

Stating the Requirement Precisely

No Nontrivial Combination Yields Zero

A collection {e₁, ..., eₖ} satisfies the independence requirement if the only scalars c¹, ..., cᵏ solving c¹e₁ + ⋯ + cᵏeₖ = 0 are c¹ = ⋯ = cᵏ = 0; if any other, nontrivial solution exists, at least one vector in the collection is redundant, expressible in terms of the others, and the collection fails the requirement.

Distinct From, but Paired With, the Span Requirement

Independence alone does not guarantee that a collection reaches every vector in the space; a collection can be independent while spanning only a proper subspace, as happens whenever fewer than the full dimension's worth of independent vectors are chosen. A basis requires both the span requirement and the independence requirement simultaneously, and a collection satisfying only independence, without spanning, is a linearly independent set rather than a basis.

Why Independence Guarantees Unique Components

Two Different Expansions Would Force a Contradiction

If a vector v admitted two different expansions in the same basis, v = c^i e_i and v = d^i e_i with some c^i ≠ d^i, subtracting the two would give (c^i − d^i)e_i = 0, a nontrivial linear combination summing to zero unless every coefficient difference is itself zero; the independence requirement is exactly what rules this out, forcing c^i = d^i for every index and thereby guaranteeing that a vector's components in a given basis are uniquely determined.

Uniqueness Is the Property Tensor Index Notation Relies On

Because a fixed basis assigns exactly one component array to each tensor once independence holds, index notation can treat T^i_j as a well-defined, specific set of numbers rather than an ambiguous label for one among several possible coefficient assignments; without the independence requirement, the very notion of "the components of T in this basis" would fail to pick out a unique array at all.

Verifying Independence in Practice

The Determinant Test in Finite Dimensions

For a candidate collection of exactly n vectors in an n-dimensional space, arranging their components (relative to any already-known basis) as the columns of a square matrix and checking that its determinant is nonzero is a standard, direct test of independence; a zero determinant signals that some nontrivial linear combination of the columns vanishes, exactly the failure condition the independence requirement excludes.

Row Reduction for a General Count of Vectors

When the number of candidate vectors is not necessarily equal to the space's dimension, independence is verified by row-reducing the matrix formed from their components and confirming that no row reduces entirely to zero (equivalently, that the matrix has full column rank), a method that applies regardless of whether the collection is being checked for independence alone or for both independence and span together.

Diagram of Independent Versus Dependent Collections

Failure of the Requirement and Its Consequences

Redundant Vectors Break the Basis System Structure

If independence fails, the induced dual basis is not well-defined in the usual sense, since the defining conditions eⁱ(eⱼ) = δⁱⱼ cannot be simultaneously satisfied for a redundant collection; the entire basis system structure — dual basis, induced tensor product bases, and the transformation rule linking them — depends on independence holding for the primal collection, exactly as it depends on the span requirement, and failure of either one prevents that structure from being correctly established.

Restoring Independence by Removing Redundant Vectors

When a candidate collection is found to violate the independence requirement, the standard remedy is to identify and discard whichever vectors are expressible in terms of the others, retaining a strictly smaller, genuinely independent subcollection; if the resulting reduced collection also satisfies the span requirement, it constitutes a valid basis, while if it spans only a subspace, further vectors must be added, independent of all those already retained, until both requirements are simultaneously met.