Canonical Gymnasium Mathematics: Sek II Linear Algebra / Matrices Audit

Snapshot: 2026-04-05

Purpose:

• review the current canonical Sek II Lineare Algebra / Matrizen inventory after the packaging passes for lower-secondary topics, analysis, stochastics, and analytic geometry / vectors
• use already reviewed upper-secondary state evidence to test whether the current canonical matrix cuts are pedagogically stable
• define the next canonical work packages for Sek II Lineare Algebra / Matrizen

Scope

In scope:

• the canonical upper-secondary matrix / linear-algebra topic surface in
• curricula/DE/Gymnasium/canonical/DE_DEU_S_GYM_CANONICAL_MATHEMATIK.de.json
• the already reviewed upper-secondary source and mapping evidence from
• BB
• BE
• BY
• HB
• HH
• HE
• MV
• NI
• NW
• RP
• SH
• SN
• ST
• TH

Out of scope:

• direct canonical JSON refactoring in this step
• learner-facing composition views
• the space/vector branch except where the current broad Q2 overview still mixes it with matrices

Reviewed source evidence

Hessen (HE)

Observed signal:

• HE remains the mature donor baseline for the upper-secondary matrix branch
• the Hessen lane confirms that the canonical graph already has real material for
• matrix representations
• matrix arithmetic
• transition processes and stochastic matrices
• fixvectors and long-term behavior
• linear geometric mappings

Bremen (HB)

Observed signal:

• HB already contributes reviewed corridors for:
• LA2-3 linear systems, matrix calculus, and redistribution models
• LA4 population matrices, fixvectors, and long-term behavior
• Bremen therefore pressure-tests whether the canonical graph separates
• matrix entry and arithmetic
• transition modelling
• stable states
• long-term development clearly enough

Hamburg (HH)

Observed signal:

• HH already contributes a reviewed Modul 7 corridor with:
• linear systems in context
• transition graphs, transition matrices, and state vectors
• matrix multiplication and inverse matrices
• fixvectors and long-term behavior
• Hamburg increases pressure on whether the current visible split between representation, arithmetic, transition modelling, and long-term interpretation is already stable enough

Sachsen (SN)

Observed signal:

• SN now adds a first reviewed upper-secondary matrix corridor directly on the broad upper snapshot
• the Sachsen Grundkurs split cleanly separates:
• matrices as a representation for linear systems
• matrix multiplication and Gauss-Jordan elimination
• geometric and network-style applications with rotations and interdependency structures
• Sachsen therefore supports the frozen LM2-LM5 surface without forcing another package-level bridge

Rheinland-Pfalz (RP)

Observed signal:

• RP now adds a first reviewed upper-secondary Raum und Form split directly on the Grundfach / Leistungsfach lane
• the Rheinland-Pfalz source explicitly separates
• A1 matrix/application work
• A2 broad geometry work on lines and planes in space
• this makes RP a useful pressure test for whether the frozen matrix surface can stay stable even when matrix work is still mixed across representation, operations, powers, and geometric mappings

Mecklenburg-Vorpommern (MV)

Observed signal:

• MV now adds a first reviewed broad upper-secondary matrix corridor on the active qualification-phase snapshot lane
• the current Mecklenburg-Vorpommern source still exposes matrix work only through one mixed Vektoren und Matrizen band
• MV therefore confirms that the frozen shared matrix surface can absorb a reviewed broad corridor without forcing a sharper visible LM2-LM5 bridge package

Thueringen (TH)

Observed signal:

• TH now adds a first reviewed broad upper-secondary matrix corridor on top of the newly activated Klassenstufe 11 / 11/12 snapshot lane
• the current Thueringen source still exposes no explicit matrix-only strip and still mixes the available matrix-relevant evidence into the broad geometry / analytic-geometry packaging
• TH therefore confirms that the frozen shared matrix surface can absorb a reviewed broad corridor even when the source currently only provides one shared Raum-/AGV-style bridge instead of a narrower explicit matrix strand

Nordrhein-Westfalen (NW)

Observed signal:

• NW already carries an explicit upper-secondary geometry / linear-algebra entry strip plus reviewed LK geometry / linear-algebra strips on the shared Sek-II space/matrix surface
• the retained mixed vector/line/LGS application leaf and the LK linear-equation-system side lane now stay on that same shared matrix/space surface instead of forcing a sharper bridge between the analytic-geometry and matrix packages

Schleswig-Holstein (SH)

Observed signal:

• SH currently exposes no explicit matrix-only strip, but its upper-secondary phase lane is fully mapped and keeps the broad area bridge Geometrie -> Raum, Matrizen und lineare Modelle (Sek II) stable on the shared upper-secondary matrix/space surface
• Schleswig-Holstein therefore resolves as a broad anchor lane for the matrix topic rather than as evidence for another shared canonical package

Sachsen-Anhalt (ST)

Observed signal:

• ST currently carries a fully mapped upper-secondary pilot snapshot with explicit GA/EA analysis, stochastics, and analytic-geometry corridors, but still no narrower explicit matrix-only strip
• Sachsen-Anhalt therefore acts as another broad anchor lane: the remaining Raum/LM residue stays on the shared upper-secondary matrix/space surface and does not force another canonical bridge package

Current canonical linear-algebra inventory

The canonical graph is already materially seeded here.

Important current package surfaces:

1. Matrizen zur Beschreibung von Übergangsprozessen
2. Matrizen als Darstellungsform verstehen
3. Übergangsprozesse mit Graphen und Matrizen beschreiben
4. Mit Matrizen rechnen
5. Stabile Zustände mithilfe von Fixvektoren bestimmen
6. Langfristige Entwicklung von Übergangsprozessen (LK)
7. the separate branch for lineare geometrische Abbildungen

Audit judgment

The canonical Sek-II linear-algebra topic is not missing a backbone. The main risk is packaging, not first missing content.

The most relevant tensions are:

1. matrix entry and transition modelling may still sit too close together in the visible surface
2. matrix arithmetic and multi-step process interpretation need an explicit shared-core versus LK-depth check
3. fixvectors and long-term behavior should stay visible as a late continuation and not be absorbed back into a single broad matrix corridor
4. the branch for linear geometric mappings needs a deliberate role decision: shared matrix topic surface or separate retained continuation

Findings

1. Matrix entry should stay visible as a separate package

The reviewed state evidence, now also including the broad Mecklenburg-Vorpommern corridor, supports a stable common entry corridor for:

• matrices as structured representations
• reading entries over rows and columns
• first state-vector interpretations

This package surface should remain explicit and should not be absorbed into later transition or long-term corridors.

2. Transition modelling is a real shared-core package

The reviewed lanes support a shared visible corridor for:

• transition graphs
• transition matrices
• state vectors
• matrix-vector interpretation in context

This package is not just a Hessen artifact. It is pressure-tested by both Bremen and Hamburg.

3. Matrix arithmetic and multi-step process reasoning need an explicit boundary check

The current graph already has material for:

• matrix operations
• matrix-vector products
• inverse matrices in simpler settings
• matrix powers for multi-step processes

The open question is whether the visible package surface separates these ideas cleanly enough from both basic representation work and the late long-term branch.

4. Fixvectors and long-term behavior should remain an explicit late continuation

The current graph already exposes late matrix surfaces for:

• fixvectors / stable states
• matrix powers
• long-term development of transition processes

That separation is useful and should be treated as materially real unless reviewed residue proves otherwise.

5. Linear geometric mappings need a conscious steady-state role

The canonical graph already has a separate matrix-based branch for linear geometric mappings. The open question is not whether this material exists, but whether it should remain a clearly distinct matrix continuation or be pulled more strongly into the general matrix package surface.

Proposed canonical work packages

For Sek II Lineare Algebra / Matrizen, use these work packages:

1. LM1 Matrix representations, entries, and first state-vector readings
2. matrices as structured representations
3. rows, columns, entries

4. first readings of matrix data in context

5. LM2 Transition processes, graphs, matrices, and state vectors

6. transition graphs
7. transition matrices

8. state vectors and matrix-vector interpretation

9. LM3 Matrix operations and multi-step process reasoning

10. matrix arithmetic
11. inverse matrices where canonically appropriate

12. matrix powers and multi-step interpretation

13. LM4 Fixvectors, stable states, and long-term development

14. fixvectors
15. stationary states

16. long-term development and advanced continuation

17. LM5 Linear geometric mappings and matrix-based transformations

18. mapping matrices
19. image points and composition as matrix products

Residue update: NI projection-with-matrices pressure test

The reviewed NI upper-secondary lane adds the currently strongest pressure test for the open LM5 question because it contains explicit matrix-based projection work rather than only transition-matrix modeling.

Key reviewed NI evidence:

1. 67bd9b91-3c08-4d20-9200-8241d66cb2c1
2. Projektionen vom Raum in die Ebene mit Matrizen beschreiben und Punktkoordinaten fuer Schraegbilder berechnen
3. maps to canonical 803d910d-96d1-5118-b9ca-29e93d0da76d
4. Projektionen im Raum untersuchen (LK)
5. partial

Judgment:

1. LM5 should remain a separate visible continuation
2. the current LM2-LM5 package cut is stable enough to freeze
3. no additional bridge package is justified between transition matrices and geometric mapping matrices
4. explicit role decision relative to the general matrix branch

Design step executed

The first canonical Sek-II linear-algebra packaging pass is now in place.

Accepted package surface:

1. LM2 is now visible as Übergangsprozesse, Zustandsvektoren und Übergangsmatrizen.
2. LM3 is now visible as Matrixoperationen und Mehrschrittprozesse.
3. LM4 is now visible as Fixvektoren, stabile Zustaende und Langzeitverhalten.
4. LM5 is now visible as Lineare geometrische Abbildungen und Abbildungsmatrizen.

Important boundary decision:

1. the former broad matrix corridor has been tightened into a summary surface over LM1-LM4
2. transition modelling is now kept visibly separate from matrix operations
3. late fixvector/long-term work remains an explicit continuation and is not reabsorbed into the operational corridor
4. the matrix-based branch for linear geometric mappings remains explicit as its own visible continuation
5. no new atomic wave was created; only visible package surfaces were clarified

First reviewed realignment judgment

The reviewed BB / BE / BY / HB / HH / HE / RP / SN pressure test does not force another canonical package split.

Observed result:

1. HB now aligns more cleanly because
2. matrix-operation residue no longer hangs on the old broad operation atom
3. transition-model residue lands on the explicit LM2 surface
4. SN now fits the frozen visible matrix surface cleanly enough on a newly opened GK corridor:
5. representation work lands on Matrizen als Darstellungsform verstehen
6. arithmetic / Gauss-Jordan work lands on LM3
7. rotation/interdependency work lands on LM5
8. fixvector and late continuation residue land on the explicit LM4 surface
9. HH now aligns more cleanly because
10. transition-modelling residue lands on LM2
11. matrix-operation and multi-step residue land on LM3
12. fixvector and long-term residue land on LM4
13. BB, BE, and BY now fit the frozen LM2-LM5 surface cleanly enough that no additional package-level bridge is needed

14. HE remains exact enough that no package-level correction is needed

15. RP now adds a first explicit GF/LF A1/A2 split in Raum und Form:

16. matrix/application work can stay on the shared upper-secondary matrix/space summary surface
17. the parallel geometry corridors remain separate enough that the matrix topic is not forced back into one mixed overview package
18. the source is still too broad to force a sharper package-level split inside LM2-LM5

Interpretation:

The revised visible matrix surface survives the first reviewed realignment pass without forcing a new shared canonical package. The remaining tension is now mainly source-granularity residue, not an obvious canonical gap.

Additional reviewed close-out evidence (NW, SH, ST)

1. NW now confirms that explicit upper-secondary geometry / linear-algebra entry strips, LK side strips, and LGS residue can all stay on the frozen LM surface without forcing a new shared package-level bridge.
2. SH confirms that a fully mapped upper-secondary lane can still remain only a broad Geometrie -> Raum, Matrizen und lineare Modelle (Sek II) anchor without leaving open matrix-package debt.
3. ST confirms the same broad-anchor outcome on a fully mapped GA/EA overview lane: lack of an explicit matrix-only strip is accepted source granularity, not a reason to reopen LM2-LM5.

Recommendation

Do not widen more bundesland mappings on Sek-II linear algebra unless another reviewed lane exposes a concrete shared gap.

Open questions now concentrated here:

1. Are LM3 and LM4 now stable enough as separate visible continuations, or will broader multi-state residue force another cut?
2. Does another reviewed lane require a clearer visible bridge between broad matrix overview parents and the now tighter late packages?

Recommended next concrete step

Freeze package churn here unless another reviewed lane exposes a concrete shared gap.

At this point, the next productive mode is:

1. residue control on additional reviewed lanes
2. or targeted reopening of the strongest remaining package tension only if real evidence forces it

Keep LM1 stable unless later reviewed residue shows that the entry package still leaks too much later transition or long-term material.

Close-out judgment (2026-04-05)

The current Sek-II linear-algebra / matrices sweep can now be treated as closed.

Why this is now strong enough:

1. the accepted visible package surface stays at LM1-LM5
2. the reviewed BB/BE/BY/HB/HH/HE/NI/NW/RP/SN corridors align without forcing another shared package-level bridge
3. the broader MV/SH/ST/TH overview lanes are now resolved as stable anchor evidence on the shared space/matrix surface rather than as open matrix debt
4. the remaining tension stays at source-granularity residue, not at a missing shared canonical matrix corridor

Exit criteria for this audit

This topic audit is complete when:

1. the canonical subpackage boundaries above are either accepted or revised
2. the canonical graph has a stable Sek-II linear-algebra packaging
3. reviewed BB, BE, BY, HB, HH, HE, NI, NW, RP, and SN evidence can be described as aligned to the revised packaging or intentionally broader because of source granularity, while the broader MV, SH, ST, and TH overview lanes are explicitly accepted as resolved anchor evidence
4. it is clear whether long-term behavior and linear geometric mappings require further visible separation in the canonical graph

Baden-Wuerttemberg upper-secondary Gauss parents connected (2026-04-04)

• the broad BW course-stage Gauss / linear-system corridor now sits on the shared Sek-II space/matrix-model surface
• the BW Basisfach: Lineare Gleichungssysteme und Stufenform parent now sits on the same broad LM surface
• the BW Leistungsfach: Gaußverfahren, Matrixschreibweise und geometrische Deutung parent also stays on that shared LM surface instead of forcing a sharper package-level bridge