Kernel of a linear map

Mathematical statement

This slice binds an exact 2-by-2 integer matrix to T:R²→R². Symbolic null equations or exact row reduction describe the full kernel. A bounded coordinate search covers only its box and cannot replace the whole real kernel. A nonzero target fiber is an affine solution set, not the kernel.

Representative formula (exact TeX)

K(T)

Assumptions

• The complete 2-by-2 matrix defines the usual real linear map on R².
• The kernel target is fixed as the zero vector and all divisions are exact rational operations.

Symbol scope

• K — linear-map kernel
• T — specified linear map

Intuition, definition, and misconceptions

Check membership by evaluating A*x=0 exactly.

Describe the complete kernel with null equations or a basis.

Failure of a bounded search does not establish global absence in R²; a nonzero target fiber is not the kernel.

Do not call a nonzero fiber A*x=b a kernel or infer the whole kernel from a finite grid.

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 and its linearity obligations are fixed.
• Real vector subspace — The kernel forms a source subspace.
• Equation — Kernel vectors solve A*x=0.

Verification boundary

Only exact 2-by-2 matrices are handled; every bounded result retains its explicit range.

not_formalized

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

Next obligation

Add nullity 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.