14.20.3 Tensor Map Product Linearity Verification

Verifying linearity of tensor map products ensures compatibility with linear algebra structures, essential for tensor algebra consistency and operations.

Tensor Map Product Linearity Verification is the stage of the verification procedure confirming that the induced assignment $t \mapsto (f \otimes g) (t)$ is additive and scalar-homogeneous on arbitrary elements of $V \otimes W$ , not merely on simple tensors, so that $f \otimes g$ genuinely qualifies as a linear map rather than only as a well-defined function.

What Distinguishes This From the Preceding Checks

Well-Definedness Is Not the Same as Linearity

Domain verification confirms that a rule proposed on pairs descends to a genuine function on $V \otimes W$ ; it does not by itself confirm that this function is linear once elements of $V \otimes W$ beyond simple tensors are considered, since a well-defined function on a vector space need not respect vector addition or scalar multiplication. Linearity verification addresses this separately.

Two Sources of Linearity to Confirm

There are two distinct places linearity can be established: linearity of the induced map when constructed through the universal property, which is automatic by construction, and linearity of a directly proposed formula, for instance a componentwise formula, which must be checked explicitly since nothing about writing down a formula guarantees it defines a linear function.

Automatic Linearity via the Universal Property

Linearity Is Built Into the Construction

When $f \otimes g$ is defined as the unique map produced by the universal property of the tensor product from the bilinear assignment $(v, w) \mapsto f (v) \otimes g (w)$ , the universal property guarantees the resulting map is linear as part of its statement; no separate check is required in this case, since the universal property produces a linear map by definition, and domain verification of the underlying bilinear assignment is the only work needed.

Why the Guarantee Still Deserves Restating

Even though this linearity is automatic, it is worth restating explicitly as a checkpoint, because subsequent steps of the overall procedure, such as comparing a candidate map $h$ to $f \otimes g$ , rely on both sides being confirmed linear before agreement on a spanning set can be used to conclude agreement everywhere.

Explicit Linearity Verification for a Component Formula

Additivity on General Coefficient Arrays

Given a candidate formula $s^{a b} = \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}$ acting on the coefficient array $t^{i j}$ of a general element $t$ , additivity requires

\sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} (t_{1}^{i j} + t_{2}^{i j}) = \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t_{1}^{i j} + \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t_{2}^{i j}

which holds because a finite double sum of scalar products distributes term by term over an entrywise sum of two arrays, a property of ordinary arithmetic on the field $K$ rather than anything special to tensor products.

Homogeneity on General Coefficient Arrays

Similarly, homogeneity requires

\sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} (α t^{i j}) = α \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}

which similarly follows by factoring the scalar $α$ out of the finite double sum, valid for every choice of output indices $a$ and $b$ simultaneously.

What Linearity Verification Does Not Establish

It Does Not Confirm the Formula Matches f⊗g

Confirming that a candidate formula is linear only shows it defines some linear map on $V \otimes W$ ; it says nothing about whether that linear map coincides with $f \otimes g$ specifically. Many linear maps besides $f \otimes g$ exist on the same space, so linearity verification must be paired with the remaining steps of the tensor map product verification procedure, particularly the check on basis input elements, before the candidate can be identified as $f \otimes g$ itself.

It Does Not Substitute for Domain Verification

A formula can be perfectly linear in the coefficient array $t^{i j}$ once that array is fixed, while still failing to be well defined if two different pairs $(v, w)$ produce the same tensor $t$ but are handled inconsistently upstream of the coefficient array; linearity verification presupposes that the coefficient array itself is already well defined, which is the separate concern of domain verification.