14.18.5 Tensor Map Product Basis Expansion Relation

The Tensor Map Product Basis Expansion Relation describes how tensor products expand basis elements through linear mappings in algebraic structures.

Tensor Map Product Basis Expansion Relation is the identity connecting the coefficients of an arbitrary element of $V \otimes W$ , expanded in basis input elements, to the coefficients of its image under $f \otimes g$ , expanded in basis output elements, obtained by extending the tensor map product basis formula from basis input elements to general elements by linearity.

The Two Expansions Being Related

Expansion of the Input

Any element $t \in V \otimes W$ has a unique expansion in the basis input elements,

t = \sum_{i} \sum_{j} t^{i j} e_{i} \otimes f_{j}

Expansion of the Output

Likewise, its image $(f \otimes g) (t)$ has a unique expansion in the basis output elements,

(f \otimes g) (t) = \sum_{a} \sum_{b} s^{a b} e_{a}' \otimes f_{b}'

The expansion relation is the formula expressing $s^{a b}$ directly in terms of the numbers $t^{i j}$ and the entries of $F$ and $G$ , without any further reference to $t$ or its image as abstract vectors.

Derivation of the Relation

Applying Linearity to the Input Expansion

Applying $f \otimes g$ to the expansion of $t$ and using linearity to move the map inside the double sum gives

(f \otimes g) (t) = \sum_{i} \sum_{j} t^{i j} (f \otimes g) (e_{i} \otimes f_{j})

Substituting the Basis Formula

Replacing each $(f \otimes g) (e_{i} \otimes f_{j})$ by the tensor map product basis formula and merging the resulting sums over $a$ and $b$ with the outer sums over $i$ and $j$ yields

(f \otimes g) (t) = \sum_{a} \sum_{b} (\sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}) e_{a}' \otimes f_{b}'

The Expansion Relation

Comparing this with the definition of $s^{a b}$ , and invoking uniqueness of expansion in the basis output elements, gives the expansion relation

s^{a b} = \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}

valid for every choice of output indices $a$ and $b$ , and depending on $t$ only through its own coefficients $t^{i j}$ .

Structural Reading of the Relation

As a Bridge Between Two Coordinate Systems

The expansion relation is the precise bridge connecting the coordinate description of an element of the domain, given by $t^{i j}$ against basis input elements, to the coordinate description of its image, given by $s^{a b}$ against basis output elements; it is this bridge that turns the abstract statement " $f \otimes g$ is linear" into a concrete numerical recipe.

As a Bilinear Form in the Two Coefficient Arrays

For fixed $F$ and $G$ , the relation exhibits $s^{a b}$ as linear in the array $t^{i j}$ , consistent with $f \otimes g$ being a linear map on $V \otimes W$ ; separately, for fixed $t$ , it is bilinear in the entries of $F$ and $G$ jointly, reflecting the underlying bilinearity of $(f, g) \mapsto f \otimes g$ as an operation on pairs of maps.

Special and Degenerate Cases

Reduction on a Basis Input Element

If $t$ is itself a single basis input element $e_{i_{0}} \otimes f_{j_{0}}$ , then $t^{i j}$ equals $1$ when $(i, j) = (i_{0}, j_{0})$ and zero otherwise, and the double sum in the expansion relation collapses to the single term $F_{i_{0}}^{a} G_{j_{0}}^{b}$ , recovering the tensor map product basis coefficient rule as the special case of the expansion relation applied to a basis input element.

Rank-One Coefficient Arrays

If $t^{i j}$ factors as $v^{i} w^{j}$ for vectors of coefficients $v$ and $w$ , the double sum in the expansion relation factors as a product of two independent single sums, one running over $i$ alone and one over $j$ alone, reproducing the simple-tensor image formula for $(f \otimes g) (v \otimes w)$ as a further special case.