The full kernel of a two-by-two map

Goal and scope

The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}.

An exact 2-by-2 real linear map and universal null equations; no bounded box replaces the kernel.

Givens and symbols

• calc

• G1 — A=[[1,2],[2,4]], T:R2 to R2, T(v)=Av, with kernel target (0,0).

• A — complete 2-by-2 matrix
• t — arbitrary real free variable

Rules and sources

• calc · prerequisite — Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations.
• mko_definition · cited — 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. · mko-linear-map-kernel

Main derivation

T\binom{x}{y}=A\binom{x}{y}

T(x,y)=(x+2y,2x+4y)

v\in\ker T

x+2y=0,\quad2x+4y=0

\left[\begin{array}{cc|c}1&2&0\\2&4&0\end{array}\right]

R_2\leftarrow R_2-2R_1:\quad\left[\begin{array}{cc|c}1&2&0\\0&0&0\end{array}\right]

x+2y=0

y=t,\quad x=-2t,\quad t\in\mathbb R

v=(-2t,t)

v=t(-2,1),\quad A(-2,1)^T=(0,0)^T

\ker T=\{t(-2,1):t\in\mathbb R\}

\ker T=\operatorname{span}\{(-2,1)\},\quad\dim\ker T=1

T(-1,1/2)

T(-1,1/2)=(0,0),\quad(-1,1/2)\notin\mathbb Z^2

T(x,y)=(1,2)

(x,y)=(1,0)+t(-2,1),\quad t\in\mathbb R

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S01, S04 — Substitution gives fixed output (1,2), not zero, distinguishing the sets.
• S07 · premise_audit · G1, S04 — This illustrates rather than replaces the universal symbolic parameterization.
• S06 · premise_audit · G1, S04, S05 — Infinitely many t values do not mean infinite nullity; dimension counts basis vectors.
• S05 · premise_audit · G1, S01, S04 — Both full solution coverage and generator substitution are needed, not merely one kernel element.
• S04 · inverse_equivalence · G1, S03 — Integer-only t would miss real vectors such as t=1/2.
• S03 · inverse_equivalence · G1, S02 — The zero row says 0=0, not y=0.
• S02 · inverse_equivalence · G1, S01 — A nonzero right side b defines another fiber, not the kernel.
• S01 · premise_audit · G1 — The columns do not reduce the proof to testing two inputs.

Error controls and negative controls

Zero target

The kernel fixes Av=0; Av=b is a separate fiber.

Free-variable domain

t ranges over R, not a bounded box or integers only.

• The zero row forces y=0. → reject — A zero row is only 0=0; y remains free.
• All kernel elements are integer multiples of (-2,1). → reject — t=1/2 gives a valid real kernel element.
• The fiber T^-1((1,2)) is also the kernel. → reject — It omits zero and has a nonzero output.

Result and limitations

The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: f6f3f4d9ab5a376fa1c72b390832230ea9415f6f3e6c3963d0c1a0cf9ee2e43a
• original source SHA-256: f6f3f4d9ab5a376fa1c72b390832230ea9415f6f3e6c3963d0c1a0cf9ee2e43a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)