2.8.1 Tensor Span Generated Set

Tensor Span Generated Set is the subspace produced as the output of the span relation applied to a chosen collection of tensors, viewed as an object in its own right rather than as the process that produces it, together with the study of how such a generated set depends on, and can be reconstructed from, various choices of generating collection. Where the span relation describes the operation of forming all linear combinations, the generated set is the resulting subspace itself, with its own dimension, its own minimal generating sets, and its own position among the subspaces of the ambient tensor space.

The Generated Set as an Object

Setting

Let $$V$$ be a vector space over a field $$F$$, and let $$S\subset T_s^r\left(V\right)$$ be any collection of tensors of type $$\left(r,s\right)$$. The generated set is the subspace $$W=\text{span}\left(S\right)$$, and $$S$$ is said to be a generating set, or a set of generators, for $$W$$.

Distinguishing the Set from Its Generators

The generated set $$W$$ is a fixed subspace, but it may arise from many different generating sets, so the identity of $$W$$ as a subspace must be distinguished from the identity of any particular $$S$$ used to describe it; two different generating sets producing the same subspace are considered equivalent for the purpose of describing $$W$$, even though they may differ as sets of tensors.

Multiple Generating Sets for the Same Generated Set

Non-Uniqueness of Generators

A single generated set $$W$$ generally admits infinitely many distinct generating sets. If $$S$$ generates $$W$$, then so does any set obtained from $$S$$ by adjoining additional elements of $$W$$, by replacing an element with a nonzero scalar multiple of itself, or by replacing an element with its sum with another element of $$S$$.

Minimal Generating Sets

Among all generating sets for a fixed $$W$$, those that are also linearly independent are the minimal generating sets, and these are precisely the bases of $$W$$ as a vector space in its own right. Every generated set admits at least one minimal generating set, obtainable from any generating set by discarding dependent elements one at a time.

Dimension of the Generated Set

Bound from a Given Generating Set

If $$S$$ is finite with $$m$$ elements, the generated set satisfies $$\text{dim}W\le m$$, with equality exactly when $$S$$ is itself linearly independent, so a minimal generating set has size exactly $$\text{dim}W$$.

Bound from the Ambient Space

Since $$W\subset T_s^r\left(V\right)$$, the generated set additionally satisfies $$\text{dim}W\le n^{r+s}$$, with equality only when $$W$$ is the entire tensor space, which occurs exactly when $$S$$ already contains a spanning set for $$T_s^r\left(V\right)$$.

Examples of Generated Sets

A Set Generated by a Single Tensor

For a single nonzero tensor $$T$$, the generated set $$\text{span}\left(T\right)=\left\{\alpha ,T,:,\alpha ,\in ,F\right\}$$ is one-dimensional, consisting of every scalar multiple of $$T$$, a line through the origin of $$T_s^r\left(V\right)$$.

The Set Generated by Simple Tensors

The set of all simple tensors of a given type does not itself form a subspace, since a sum of two simple tensors is generally not simple, but the set it generates is the entire tensor space $$T_s^r\left(V\right)$$, since the basis tensor products, which are themselves simple, already span this space.

The Symmetric and Antisymmetric Subspaces as Generated Sets

The subspace of symmetric tensors is the set generated by symmetrizing every basis tensor product, and the subspace of antisymmetric tensors is the set generated by antisymmetrizing every basis tensor product; both are proper subspaces of $$T_s^r\left(V\right)$$ whenever $$r+s\ge 2$$, illustrating that a generated set need not be the whole tensor space even when its generators are drawn from a spanning family of the whole space.

The Generated Set of the Full Tensor Basis

Recovering the Whole Space

When $$S=B$$, the set of all induced basis tensor products, the generated set is the entire tensor space, and this particular case is the tensor basis span structure: $$B$$ is not merely a generating set but a minimal one, so the generated set achieves the maximum possible dimension $$n^{r+s}$$ using the fewest possible generators.

Generated Sets and the Lattice of Subspaces

Containment Relations

Generated sets are partially ordered by inclusion, mirroring the monotonicity of the span relation: if $$S_1\subset S_2$$, the generated set of $$S_1$$ is contained in the generated set of $$S_2$$. This containment structure organizes the subspaces of $$T_s^r\left(V\right)$$ generated by various collections into a layered hierarchy, ranging from the zero subspace, generated by the empty set, up to the full tensor space, generated by any spanning set.

Intersections and Sums of Generated Sets

Given two generated sets $$W_1=\text{span}\left(S_1\right)$$ and $$W_2=\text{span}\left(S_2\right)$$, their sum $$W_1+W_2$$ is itself the generated set of $$S_1\cup S_2$$, while their intersection $$W_1\cap W_2$$ is itself a subspace, and hence itself a generated set, though generally not describable simply as the span of the intersection of the original generating sets.