Image of a linear map

Mathematical statement

This slice uses exact 2-by-2 matrices and symbolic column-span certificates. Membership is exact real consistency of A*x=y. A bounded fiber search supplies witnesses or bounded nonexistence only, not global nonmembership in the real image. The image must not be conflated with the entire declared codomain or a sampled output list.

Representative formula (exact TeX)

I(T)

Assumptions

• The complete 2-by-2 matrix defines T:R²→R².
• Global membership uses a symbolic linear system; bounded search has a separate explicit boundary.

Symbol scope

• I — linear-map image
• T — specified linear map

Intuition, definition, and misconceptions

The image is the full real span of the matrix columns.

Membership is equivalent to existence of a real solution of A*x=y.

Failure of bounded search is not global nonmembership.

The image need not equal the codomain, and finitely many sampled inputs do not establish the complete output range.

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 map between real vector spaces — The map T is fixed.
• Real vector subspace — The image is a subspace of the target.
• Function mapping — Image and codomain have different roles.

Verification boundary

Only exact 2-by-2 matrices and explicit symbolic certificates are handled.

not_formalized

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

Next obligation

Add rank and rank-nullity.

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.