14.18.2 Tensor Map Product Basis Output Element

The Tensor Map Product Basis Output Element shows how tensor maps produce basis elements in product spaces through linear combinations.

Tensor Map Product Basis Output Element is the simple tensor $e_{a}' \otimes f_{b}'$ , built from one basis vector of the codomain $V^{'}$ and one basis vector of the codomain $W^{'}$ , in terms of which the image of any basis input element under $f \otimes g$ is expanded, and which itself spans the codomain $V^{'} \otimes W^{'}$ exactly as the basis input elements span the domain.

Definition and Setting

Construction From the Two Codomain Bases

Given a basis ${e_{1}', \dots, e_{p}'}$ of $V^{'}$ and a basis ${f_{1}', \dots, f_{q}'}$ of $W^{'}$ , each basis output element is

e_{a}' \otimes f_{b}' \in V^{'} \otimes W^{'}, 1 \leq a \leq p, 1 \leq b \leq q

giving exactly $p q$ such elements, one for each ordered pair $(a, b)$ , and this collection forms a basis of $V^{'} \otimes W^{'}$ .

Role as the Target of the Basis Formula

The basis output elements are the fixed reference frame against which the image of every basis input element is measured: rather than describing $(f \otimes g) (e_{i} \otimes f_{j})$ in the abstract, the basis formula expresses it as an explicit linear combination of these particular elements, with coefficients given by the tensor map product basis coefficient rule.

The Image of a Basis Input Element Is a Combination of Basis Output Elements

Expansion Formula

For each basis input element $e_{i} \otimes f_{j}$ , the tensor map product basis formula gives

(f \otimes g) (e_{i} \otimes f_{j}) = \sum_{a = 1}^{p} \sum_{b = 1}^{q} F_{i}^{a} G_{j}^{b} e_{a}' \otimes f_{b}'

a sum ranging over all $p q$ basis output elements, generally with more than one nonzero coefficient.

The Image Is Usually Not Itself a Basis Output Element

Even though $f (e_{i}) \otimes g (f_{j})$ is a simple tensor of two vectors of the codomains, it coincides with a single basis output element only in the special case where $f (e_{i})$ is itself one basis vector $e_{a}'$ and $g (f_{j})$ is itself one basis vector $f_{b}'$ , with all other coefficients zero; in general the images of basis vectors under $f$ and $g$ are nontrivial linear combinations, so the image of a basis input element spreads across several basis output elements at once.

Reading Off Coefficients Against the Output Basis

The Coefficient of a Fixed Output Element

Because the basis output elements are linearly independent, the coefficient of one particular basis output element $e_{a}' \otimes f_{b}'$ in the expansion of $(f \otimes g) (e_{i} \otimes f_{j})$ is well defined and unambiguous, namely $F_{i}^{a} G_{j}^{b}$ , and this well-definedness is exactly what allows the basis output elements to serve as the rows of the matrix representing $f \otimes g$ .

Dual Description via Coordinate Functionals

Equivalently, if ${e^{a}}$ and ${f^{b}}$ denote the dual coordinate functionals picking out the coefficient of $e_{a}'$ and of $f_{b}'$ respectively, the coefficient of the basis output element $e_{a}' \otimes f_{b}'$ inside any tensor $s$ of the codomain is $(e^{a} \otimes f^{b}) (s)$ , giving a coordinate-free way to extract exactly the number appearing next to each basis output element.

Relation to Basis Input Elements

Structural Parallel

The basis output elements of $V^{'} \otimes W^{'}$ play exactly the same structural role for the codomain that the basis input elements play for the domain $V \otimes W$ : both are formed by pairing one basis vector from each tensor factor, both form a basis of the corresponding tensor product space, and both index the rows and columns of the matrix representing $f \otimes g$ , with basis input elements indexing the columns and basis output elements indexing the rows.

Asymmetry of the Map Between Them

Despite this structural parallel, the map itself is not symmetric between the two sets: a single basis input element maps, in general, to many basis output elements simultaneously with nonzero coefficients, while conversely a single basis output element can appear with a nonzero coefficient in the images of many different basis input elements, reflecting the fact that $f \otimes g$ need not be diagonal, or even square, when the domain and codomain dimensions differ.