2.14.2 Tensor Linear Independence Basis Relation

Tensor Linear Independence Basis Relation is the connection between linear independence and the definition of a basis, expressed by the fact that a collection of vectors forms a basis of a vector space precisely when it is both linearly independent and spans the entire space, so independence supplies the non-redundancy half of what a basis requires. This relation clarifies why independence alone, without spanning, is insufficient to guarantee a basis, and why the two conditions must be verified together.

Formal Statement

Basis as Independence Combined With Spanning

A collection of vectors qualifies as a basis of a vector space exactly when it is linearly independent and its span equals the entire vector space.

Neither Condition Alone Suffices

A set can be linearly independent without spanning the space, in which case it is merely a partial basis of a proper subspace, and a set can span the space without being independent, in which case it is a spanning set containing redundant vectors.

Building a Basis From an Independent Set

Extension to a Full Basis

If a linearly independent collection of vectors does not yet span the entire vector space, it can be extended by adding further vectors, one at a time, chosen outside the current span, while preserving independence, until the resulting collection spans the space and becomes a basis.

Reduction From a Spanning Set

Conversely, if a spanning set is not independent, redundant vectors can be identified and removed one at a time without shrinking the span, eventually arriving at a linearly independent spanning set, which is a basis.

Maximality and Minimality Characterizations

Basis as a Maximal Independent Set

A basis can be equivalently characterized as a maximal linearly independent collection of vectors, meaning no additional vector from the space can be added without destroying independence.

Basis as a Minimal Spanning Set

A basis can also be equivalently characterized as a minimal spanning collection, meaning no vector can be removed from it without causing the remaining vectors to fail to span the entire space.

Role in Tensor Construction

Confirming Suitability of Basis Vectors for Coordinates

The basis relation is what justifies checking both independence and spanning before accepting a proposed collection of vectors as the basis used to assign coordinates to vectors feeding into a tensor construction.

Consistency Across Tensor Factor Spaces

When multiple vector spaces each require their own basis to support tensor construction, the same independence-and-spanning relation must be verified separately in every factor space, ensuring each factor contributes a genuinely valid coordinate system.

Summary of Key Properties

Two Conditions United Into One Definition

Tensor Linear Independence Basis Relation shows that a basis is defined by the simultaneous satisfaction of independence and spanning, neither of which alone is sufficient.

Bidirectional Path Between Partial Structures and a Full Basis

This relation provides constructive procedures, extension from an independent set and reduction from a spanning set, for arriving at a genuine basis starting from either partial structure.