Linear independence and bases

Mathematical statement

A finite vector family is linearly independent if a weighted sum equal to zero requires every coefficient to be zero. To be a basis of a space V, it must be independent and have span equal to V. Neither obligation replaces the other.

Representative formula (exact TeX)

B(V)

Assumptions

• Exercises use finite integer-coordinate vector families, with rank computed by exact rational elimination.
• Determinants and coordinate uniqueness are used only for ordered square R² bases, not extended to non-square families.

Symbol scope

• B — basis-audit record
• V — declared vector space

Intuition, definition, and misconceptions

Independence asks whether the zero vector has only the trivial coefficient representation.

Being a basis is the conjunction of independence and spanning.

Only after an ordered basis has been established does every R² target have unique coordinates in it.

Do not call a family a basis merely because it spans or merely because it is independent.

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

• Linear combinations and span — A basis must span the declared space.
• Linear systems in two variables and degenerate constraints — Independence and coordinates reduce to homogeneous or inhomogeneous systems.
• Determinants of real 2×2 matrices — A nonzero determinant checks independence and spanning for a square R² family.

Verification boundary

Rank and determinant conclusions are restricted to the explicitly declared finite coordinate model.

not_formalized

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

Next obligation

Add dimension, change of basis and matrices of linear maps.

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.