14.20.2 Tensor Map Product Codomain Verification

Tensor Map Product Codomain Verification is the stage of the verification procedure concerned with confirming that the stated codomain $V^{\prime }\otimes W^{\prime }$ of $f\otimes g$ is correct and that the image of $f\otimes g$ inside that codomain is exactly what is claimed, rather than a proper subspace or a mismatched space.

Membership of Outputs in the Stated Codomain

Immediate Membership on Simple Tensors

Since $f:V\to V^{\prime }$ and $g:W\to W^{\prime }$ by hypothesis, the value $f\left(v\right)\otimes g\left(w\right)$ is automatically an element of $V^{\prime }\otimes W^{\prime }$, so membership of individual simple-tensor outputs in the stated codomain requires no separate argument beyond confirming that $f$ and $g$ genuinely have the codomains claimed for them.

Extension to General Elements

Because $V^{\prime }\otimes W^{\prime }$ is a vector space and hence closed under linear combination, sums of such simple-tensor outputs also lie in the stated codomain, so once membership is confirmed on simple tensors, it holds automatically for the image of every element of $V\otimes W$ under $f\otimes g$.

Verifying the Exact Image

The Image Factorization Claim

The substantive part of codomain verification is confirming the identity

where the right-hand side denotes the subspace of $V^{\prime }\otimes W^{\prime }$ spanned by simple tensors of an image vector of $f$ and an image vector of $g$. This identity is not automatic from the definition alone and must be checked in both directions.

Containment in One Direction

Every basis output element appearing with a nonzero coefficient in $(f\otimes g)\left(t\right)$, for any $t$, arises as a linear combination of terms $f\left(e_i\right)\otimes g\left(f_j\right)$, each of which lies in $\text{im}\left(f\right)\otimes \text{im}\left(g\right)$, giving the containment $\text{im}(f\otimes g)\subseteq \text{im}\left(f\right)\otimes \text{im}\left(g\right)$.

Containment in the Other Direction

Conversely, any element of $\text{im}\left(f\right)\otimes \text{im}\left(g\right)$ is a linear combination of simple tensors $v^{\prime }\otimes w^{\prime }$ with $v^{\prime }=f\left(v\right)$ and $w^{\prime }=g\left(w\right)$ for some $v\in V$ and $w\in W$; each such simple tensor equals $(f\otimes g)(v\otimes w)$, giving the reverse containment and completing the equality.

Consequence for Rank as a Codomain Check

Rank Consistency

The dimension of the image, verified above to equal $\text{im}\left(f\right)\otimes \text{im}\left(g\right)$, has dimension $\text{rank}\left(f\right)\cdot \text{rank}\left(g\right)$, providing a numerical cross-check on codomain verification: computing $\text{rank}(f\otimes g)$ directly from a candidate matrix and confirming it equals this product is a quick way to detect an error in the codomain identification without redoing the full containment argument.

Detecting an Overstated Codomain

If a candidate construction claims $f\otimes g$ to be surjective onto $V^{\prime }\otimes W^{\prime }$ whenever $f$ and $g$ are surjective, codomain verification confirms this directly from the image factorization: if $\text{im}\left(f\right)=V^{\prime }$ and $\text{im}\left(g\right)=W^{\prime }$, then $\text{im}(f\otimes g)=V^{\prime }\otimes W^{\prime }$, matching the full stated codomain exactly.

Corestriction and Its Compatibility

Restricting the Codomain to the Image

If $f$ and $g$ are corestricted to their own images, producing surjective maps $f^{\prime }:V\to \text{im}\left(f\right)$ and $g^{\prime }:W\to \text{im}\left(g\right)$, codomain verification confirms that $f^{\prime }\otimes g^{\prime }$ agrees with $f\otimes g$ after the codomain of the latter is likewise corestricted to $\text{im}\left(f\right)\otimes \text{im}\left(g\right)$, since corestriction changes only the stated codomain of a map and not its values.