Span

A precise treatment of linear spans, membership, generating sets, and minimal bases.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

The span of vectors $v_1,\ldots,v_k$ is the set of all linear combinations $\operatorname{span}\{v_i\}=\{c_1v_1+\cdots+c_kv_k:c_i\in\mathbb F\}$. It is the smallest subspace containing every listed vector.

Notation and mathematical language

To test whether $b$ lies in the span of matrix columns, solve $Ac=b$. Consistency gives coefficients; inconsistency proves nonmembership. A spanning list may be dependent and nonunique, while a basis is a minimal independent spanning list.

Conceptual picture

Varying coefficients sweeps every attainable direction. Adding a vector already in the span changes the generating list but not the subspace. Span describes reachability, not just a picture of the listed arrows.

Fully worked example

Interpretation and application

Span identifies model spaces, controllable states, feature spaces, and solution families. A target near but outside a span motivates projection and least squares rather than declaring exact membership.