# The full image from consistency

[← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-linear-map-image/)

This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-linear-map-image-01 · ocme-new142-reader-adoption-20261021



## 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

S01 · impliesBefore

A\binom{x}{y}

After

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

Conditions: none

Rule: calc · Uses: G1

Leaf basis: calc

Every output is a real column combination, giving inclusion in the column span.



S02 · impliesBefore

z=xc_1+yc_2

After

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

Conditions: none

Rule: calc · Uses: G1, S01

Leaf basis: calc

Any column combination is obtained by using its coefficients as input, proving reverse inclusion.



S03 · impliesBefore

c_2=2c_1

After

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

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

The columns are proportional, reducing every output to parameter s=x+2y.



S04 · impliesBefore

s\in\mathbb R

After

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

Conditions: none

Rule: mko_definition · Uses: G1, S03

Leaf basis: mko_definition

A universal preimage shows every vector of this form is reached.



S05 · equivalentBefore

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

After

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

Conditions: none

Rule: calc · Uses: G1, S01, S04

Leaf basis: calc

Subtracting twice equation one gives necessity; if v=2u, x=u,y=0 gives sufficiency.



S06 · impliesBefore

(u,v)=(3,6)

After

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

Conditions: none

Rule: calc · Uses: G1, S05

Leaf basis: calc

Consistency holds with exact preimage (3,0), certifying membership.



S07 · impliesBefore

(u,v)=(0,1)

After

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

Conditions: none

Rule: calc · Uses: G1, S05

Leaf basis: calc

Global consistency fails, proving nonmembership rather than just absence in a box.



S08 · impliesBefore

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

After

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

Conditions: none

Rule: calc · Uses: G1, S04, S05

Leaf basis: calc

The small box misses (100,200), yet it has a global preimage, illustrating the search boundary.







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

Box boundaryA 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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