Linear combinations and span

Mathematical statement

For a vector family in a real vector space, a finite linear combination is a finite sum weighted by real coefficients. Its span is the set of all finite-length linear combinations and does not include undefined infinite series.

Representative formula (exact TeX)

S(V)

Assumptions

• Exercises use integer coordinate vectors in a fixed dimension and integer or exact rational coefficients.
• Each question audits only its explicitly supplied finite generator family and candidate witness; a fixed arity does not exhaust the abstract definition of span.

Symbol scope

• S — span operator
• V — finite generator family

Intuition, definition, and misconceptions

A weighted sum requires matching vector dimensions and the correct number of coefficients.

Supplied coefficients can reconstruct a target directly, providing a membership witness.

Two independent generators in R² permit unique coordinates to be found using their nonzero determinant.

Do not call an infinite sum a finite linear combination or infer membership in a span from dimension alone.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real vector space — Linear combinations use vector addition and real scalar multiplication.
• Linear systems in two variables and degenerate constraints — Coordinate membership can be expressed as a linear system.

Verification boundary

Only finite coordinate witnesses are audited; a general span still quantifies over all finite combinations.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add linear independence, bases and dimension.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.