14.18.1 Tensor Map Product Basis Input Element

The Tensor Map Product Basis Input Element defines how basis vectors are transformed under tensor maps, establishing foundational input for tensor algebra operations.

Tensor Map Product Basis Input Element is the specific simple tensor $e_{i} \otimes f_{j}$ , built from one basis vector of $V$ and one basis vector of $W$ , that serves as the fixed input on which the tensor map product basis formula for $f \otimes g$ is stated, and it is precisely because these elements form a basis of $V \otimes W$ that specifying the formula on them is sufficient to determine the entire map.

Identifying the Basis Input Element

Construction From Two Factor Bases

Given a basis ${e_{1}, \dots, e_{n}}$ of $V$ and a basis ${f_{1}, \dots, f_{m}}$ of $W$ , each basis input element is formed by pairing exactly one basis vector from each factor and joining them with the tensor symbol,

e_{i} \otimes f_{j} \in V \otimes W, 1 \leq i \leq n, 1 \leq j \leq m

so that there are exactly $n m$ distinct basis input elements, one for each ordered pair of indices $(i, j)$ .

Distinction From a General Simple Tensor

A general simple tensor $v \otimes w$ is formed from an arbitrary vector $v \in V$ and an arbitrary vector $w \in W$ , whereas a basis input element restricts both factors to a single basis vector each. Every basis input element is a simple tensor, but a simple tensor $v \otimes w$ is a basis input element only in the special case where $v$ and $w$ are themselves chosen basis vectors rather than arbitrary linear combinations of them.

Why the Basis Input Elements Suffice

The Set of Basis Input Elements Is a Basis

The collection ${e_{i} \otimes f_{j} ∣ 1 \leq i \leq n, 1 \leq j \leq m}$ is linearly independent and spans $V \otimes W$ , so it forms a basis of that space, of dimension

dim (V \otimes W) = dim (V) \cdot dim (W) = n m

This is not automatic from the definition of the tensor product alone; it is a theorem that must be established, and it is exactly this theorem that licenses treating $e_{i} \otimes f_{j}$ as basis input elements in the first place.

Every Element Decomposes Uniquely

Because the basis input elements form a basis, every $t \in V \otimes W$ has a unique expansion

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

with scalar coefficients $t^{i j}$ determined uniquely by $t$ , so that knowledge of how a linear map behaves on each basis input element, together with linearity, determines the map's action on every element of $V \otimes W$ without exception.

Prescribing the Map on Basis Input Elements Determines It Everywhere

Given any assignment of a target vector to each basis input element, there is exactly one linear map $V \otimes W \to V^{'} \otimes W^{'}$ agreeing with that assignment, obtained by extending linearly according to the unique decomposition above. The tensor map product basis formula is exactly such an assignment, sending $e_{i} \otimes f_{j}$ to $f (e_{i}) \otimes g (f_{j})$ , and the basis input element is precisely the object on which that assignment is made.

Role Within the Basis Formula

The Basis Input Element as Domain of the Rule

The rule

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

takes a basis input element as its argument on the left-hand side and produces, on the right-hand side, a simple tensor of the codomain that is generally not itself a basis input element of $V^{'} \otimes W^{'}$ , since $f (e_{i})$ and $g (f_{j})$ are, in general, linear combinations of the target basis vectors rather than single target basis vectors.

Indexing Consistency

The two indices $i$ and $j$ carried by a basis input element are independent of one another, ranging respectively over the index set of the basis of $V$ and the index set of the basis of $W$ ; this independence is what allows the images $f (e_{i})$ and $g (f_{j})$ to be expanded separately, using separate sums over separate target indices $a$ and $b$ , before being recombined by the tensor symbol into the final expression of the basis formula.