Matrix-represented linear maps on the real plane

Mathematical statement

A map T:R²→R² is linear exactly when T(u+v)=T(u)+T(v) and T(cv)=cT(v) for all vectors u,v and real scalars c. Here T(v)=Mv, with a 2×2 real matrix M and the column-vector convention. The affine map F(v)=Mv+b on the whole real plane is linear exactly when b is the zero vector.

Representative formula (exact TeX)

T(v)

Assumptions

• Declare the whole real plane as domain and codomain, column vectors, and a 2×2 matrix. Do not generalize these instances into finite-sample tests of arbitrary maps or dimensions.
• Exercise matrices, vectors, and scalars are safe integers, evaluated exactly.
• Zero matrices and zero vectors are valid. A linear map need not be invertible, and neither slope nor determinant is required to be nonzero.

Symbol scope

• T — matrix-represented linear map on the real plane
• v — two-component real column vector

Intuition, definition, and misconceptions

Each component of Mv is the sum of corresponding products in one row of M with v, not an entrywise product retaining a matrix output. Additivity and scalar compatibility quantify over all permitted inputs; a few samples cannot establish them for an arbitrary function.

The matrix representation specifies an explicit linear model. The exercises compute actual values on both sides of an identity, without repackaging these finite computations as a universal proof.

If b≠0, then F(0)=b≠0, so F is not linear. At b=0 it is the matrix map Mv. Changing irrelevant entries of M does not create a new affine-eligibility instance; invertibility, injectivity, and isometry have separate conditions.

Common errors include calling an affine expression with a nonzero intercept linear, or using a finite set of sample inputs to establish linearity of an arbitrary function.

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

• Finite matrices: dimensions, indexing, and transpose — Fix the 2×2 matrix and its indices.
• Plane vectors in a fixed basis — Use addition and scalar multiplication of two-component vectors.
• Function mapping — Declare inputs and outputs of the total function.

Verification boundary

Exact 2×2 integer instances can be checked; general real linearity and the matrix-to-LinearMap bridge have no newly attached formal Evidence.

not_formalized

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

Next obligation

Bridge Matrix.mulVecLin and the standard basis to addition and scalar compatibility on R²; this is not differential linearization.

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.