The full image from consistency

Goal and scope

im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not.

The entire real image of an exact 2-by-2 system, not a finite sampled output list.

Givens and symbols

• calc

• G1 — T:R2 to R2, A=[[1,2],[2,4]], c1=(1,2), c2=(2,4).

• c_1,c_2 — matrix columns
• u,v — arbitrary target coordinates

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• 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 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. · mko-linear-map-image

Main derivation

A\binom{x}{y}

x\binom{1}{2}+y\binom{2}{4}

z=xc_1+yc_2

z=T(x,y)\Rightarrow\operatorname{span}\{c_1,c_2\}\subseteq\operatorname{im}T

c_2=2c_1

T(x,y)=(x+2y)c_1

s\in\mathbb R

T(s,0)=(s,2s),\quad\operatorname{im}T=\{(s,2s):s\in\mathbb R\}

A\binom{x}{y}=\binom{u}{v}

x+2y=u,\quad 2x+4y=v\iff v=2u

(u,v)=(3,6)

6=2\cdot3,\quad T(3,0)=(3,6)

(u,v)=(0,1)

1\ne2\cdot0\Rightarrow(0,1)\notin\operatorname{im}T

|x|,|y|\le1,\quad x,y\in\mathbb Z

-3\le x+2y\le3,\quad(100,200)=T(100,0)\in\operatorname{im}T

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S04, S05 — Bounded nonexistence cannot replace S05's symbolic criterion.
• S07 · premise_audit · G1, S05 — The contradiction 1=0 excludes all real inputs.
• S06 · premise_audit · G1, S05 — The preimage is not unique; adding t(-2,1) preserves output (3,6).
• S05 · inverse_equivalence · G1, S01, S04 — Both directions are proved without a bounded no-hit inference.
• S04 · premise_audit · G1, S03 — The image is a line inside R2, not the entire declared codomain.
• S03 · premise_audit · G1, S01, S02 — s can be any real because x=s,y=0 is available.
• S02 · premise_audit · G1, S01 — Both inclusions give the full image, not merely a lower bound.
• S01 · premise_audit · G1 — x,y range over all reals, not merely the columns themselves.

Error controls and negative controls

Image and codomain

Use Ax=y consistency rather than guessing surjectivity from T:R2 to R2.

Box boundary

A finite search's negative result applies only to its declared input box.

• The codomain R2 equals the image. → reject — (0,1) fails v=2u.
• The two columns are all image elements. → reject — Every real multiple s(1,2) is an output, such as (3,6).
• Missing (100,200) in a small box proves nonmembership. → reject — (100,0) is an explicit global preimage.

Result and limitations

im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: b01b892b391c388da6d5e1fa97da0a688e710e364c061e305952d03b4ff25546
• original source SHA-256: 841d6adb1aa0065b3f001c0945ad4d8276511c210c6df28756fc85df02fed2aa
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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