2.8.5 Tensor Span Coordinate Reach

Tensor Span Coordinate Reach explores how tensor spans define reachable coordinate spaces through algebraic structures and linear combinations in multilinear algebra.

Tensor Span Coordinate Reach is the property describing which coordinate tuples over the field can actually be realized as the coordinates of some vector once a basis has been fixed, and it establishes that when the basis spans the entire vector space, every possible tuple of coefficients corresponds to a genuine vector reachable through the span. Coordinate reach connects the abstract spanning behavior of a basis to the concrete arithmetic of coordinate tuples used throughout tensor computation.

Formal Statement

Coordinate Map Induced by a Basis

Once a basis of size equal to the dimension of the space is fixed, every tuple of field elements can be mapped to a vector by forming the corresponding linear combination.

φ : F^{n} \to V, φ (c_{1}, \dots, c_{n}) = \sum_{i = 1}^{n} c_{i} b_{i}

Reach Equals the Span

The image of this coordinate map, meaning the set of all vectors reachable by varying the tuple of coefficients across the whole coefficient space, is exactly the span of the basis vectors.

image (φ) = span ({b_{1}, \dots, b_{n}})

Full Reach Under Basis Coverage

Surjectivity When the Basis Spans the Space

When the generating vectors form a basis that covers the entire vector space, the coordinate map becomes surjective, meaning every vector in the space, without exception, is the image of at least one coordinate tuple.

Partial Reach for Incomplete Spanning Sets

If the chosen vectors span only a proper subspace, the coordinate reach is limited to that subspace, and any vector lying outside it has no coordinate tuple that produces it under this coordinate map, regardless of how the coefficients are chosen.

Reach Combined With Independence

Bijectivity Under Full Basis Conditions

When the spanning vectors are also linearly independent, the coordinate map is not just surjective but bijective, so coordinate reach is paired with uniqueness, meaning each vector in the space corresponds to exactly one coordinate tuple.

Consequence for Tensor Component Assignment

This bijective reach is what permits tensor components to be assigned unambiguously once a basis is fixed, since every relevant vector has one and only one coordinate tuple to serve as its component description.

Role in Tensor Basis Span Structure

Quantifying What the Span Structure Can Represent

Within the tensor basis span structure, coordinate reach quantifies precisely which vectors are representable through coordinates relative to the current basis, giving a computational counterpart to the more abstract span subspace relation.

Bridge to Coordinate Vector Representation

Coordinate reach forms the conceptual bridge to coordinate vector representation, since it is the reach property that justifies treating a vector's coordinate tuple as a faithful stand-in for the vector itself within tensor computations.

Summary of Key Properties

Reach as the Practical Face of Spanning

Tensor Span Coordinate Reach translates the abstract notion of a spanning set into a concrete statement about which coefficient tuples produce which vectors, making span behavior directly usable in computation.

Dependence on the Chosen Generating Set

The extent of coordinate reach depends entirely on the generating set in use, so enlarging or shrinking that set correspondingly enlarges or shrinks the collection of vectors reachable through coordinates.