2.4.2 Tensor Vector Element Equality

Tensor Vector Element Equality explores when tensor components match vector elements, defining equivalence in algebraic structures through indexed comparisons.

Tensor Vector Element Equality is the equivalence relation defined on a tensor space by the identity criterion, together with the formal properties, reflexivity, symmetry, transitivity, and compatibility with the vector space operations, that make this relation behave consistently throughout tensor algebra. Where the identity criterion supplies the test for whether two tensors are the same object, equality is the relational structure built from that test, governing how substitution, simplification, and algebraic manipulation of tensors are allowed to proceed.

Equality as an Equivalence Relation

Setting

Let $V$ be a finite-dimensional vector space over a field $F$ , and consider the relation $=$ on $T_{s}^{r} (V)$ defined by agreement on every input, as established by the identity criterion. This relation satisfies the three defining properties of an equivalence relation.

Reflexivity

Every tensor equals itself:

T = T

since a multilinear map trivially agrees with itself on every input.

Symmetry

If one tensor equals a second, the second equals the first:

S = T \Rightarrow T = S

because agreement of scalar outputs is itself a symmetric condition.

Transitivity

If a first tensor equals a second, and the second equals a third, the first equals the third:

R = S and S = T \Rightarrow R = T

since agreement with a common third map on every input forces agreement between the two outer maps.

Together these three properties confirm that equality partitions descriptions of tensors into classes, each class corresponding to exactly one element of the tensor space.

Compatibility with the Vector Space Operations

Congruence with Addition

Equality respects addition: if $S_{1} = S_{2}$ and $T_{1} = T_{2}$ , then

S_{1} + T_{1} = S_{2} + T_{2}

so equal tensors may always be substituted for one another inside a sum without changing the result.

Congruence with Scalar Action

Equality likewise respects the scalar action: if $S = T$ , then for every $α \in F$ ,

α S = α T

These two congruence properties are what justify treating equality as a substitution rule throughout any calculation involving tensor addition and scalar action, not merely as a static label attached to a pair of tensors.

The Extensionality Principle

Statement

Two tensors are equal if and only if they cannot be distinguished by any input to the multilinear map:

S = T \Leftrightarrow \forall ω^{1}, \dots, v_{s} : S (ω^{1}, \dots, v_{s}) = T (ω^{1}, \dots, v_{s})

Consequence for Verification

This principle means that proving two tensor expressions equal never requires manipulating them as abstract symbols; it always suffices to evaluate both sides on an arbitrary input, or equivalently on a spanning set of basis inputs, and check that the results coincide.

Equality via Components

Finite Verification

Extensionality, applied only to basis vectors and dual basis covectors, shows that equality reduces to a finite check on components:

S = T \Leftrightarrow S_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}} = T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}

Basis Choice Does Not Affect the Outcome

Any basis may be used for this check, since the transformation law relating components across bases is linear and applies identically to both sides of the equality; agreement of components in one basis logically forces agreement of components in every other basis.

Equality Versus Weaker Notions of Sameness

Equality Versus Proportionality

Two tensors may be proportional, meaning $S = α T$ for some nonzero scalar $α$ , without being equal. Proportionality is a weaker relation than equality; it coincides with equality only in the special case $α = 1$ .

Equality Versus Equivalence Under Symmetry

In spaces of symmetric or antisymmetric tensors, permuting the factors of a simple tensor may or may not produce an equal element, depending on the symmetry type. Equality remains the strict relation requiring full agreement of the multilinear map; symmetry properties describe an additional structural relationship among specific rearrangements, not a relaxation of the equality relation itself.

Role in Algebraic Manipulation

Justifying Simplification Steps

Every simplification performed on a tensor expression, factoring a scalar out of a sum, regrouping terms, or replacing a subexpression with an equal tensor, relies on equality being a congruence with respect to addition and scalar action. Without this compatibility, algebraic manipulation of tensor expressions would not be guaranteed to preserve the underlying element being described.