Module 4: Linear Algebra for CS: Mistake Clinic

This clinic turns wrong moves into reusable judgment. Use it after each practice page and again before the quiz or checkpoint.

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Module-Specific Mistake Radar

Start with these traps. Replace or extend them with real mistakes from your own work.

Mistake to look for	Where it shows up	Symptom	Repair evidence
Finishing Elimination and Subspace Diagnostics Lab with only a final answer	Elimination and Subspace Diagnostics Lab	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Basis, Rank, and Transformation Workshop with only a final answer	Basis, Rank, and Transformation Workshop	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Orthogonality, Least Squares, and Spectral Clinic with only a final answer	Orthogonality, Least Squares, and Spectral Clinic	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Code Katas with only a final answer	Code Katas	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Treating Elimination Turns Geometry into an Algorithm as vocabulary instead of a tool	Elimination Turns Geometry into an Algorithm	The explanation names the concept but cannot decide between two cases.	Write one example, one non-example, and the rule that separates them.
Treating Matrix Multiplication and Factorization Encode Composition as vocabulary instead of a tool	Matrix Multiplication and Factorization Encode Composition	The explanation names the concept but cannot decide between two cases.	Write one example, one non-example, and the rule that separates them.

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Practice Mistake Checks

Pull any miss from these checks into your mistake log.

Elimination and Subspace Diagnostics Lab

Source: practice/01-elimination-and-subspace-diagnostics-lab.md

Identify the mistake in each claim:

1. "If two systems look different after elimination, they have different solutions."
2. "Any plane in R^3 is a subspace."
3. "The pivot columns of the reduced matrix are a basis for the original column space."
4. "If a homogeneous system has one free variable, it still only has the zero solution."

Basis, Rank, and Transformation Workshop

Source: practice/02-basis-rank-and-transformation-workshop.md

Find the error in each statement:

1. "Three vectors in R^3 always form a basis."
2. "If two matrices differ, they must represent different transformations."
3. "A linear map can include a constant offset as long as the matrix part is linear."
4. "Dimension equals the number of equations used to describe the space."

Orthogonality, Least Squares, and Spectral Clinic

Source: practice/03-orthogonality-least-squares-and-spectral-clinic.md

Find the error in each statement:

1. "Least squares finds an exact solution that the original system hid."
2. "Orthogonal vectors are only a two-dimensional picture idea."
3. "Every matrix with repeated eigenvalues is diagonalizable."
4. "Singular values are just the eigenvalues written differently."

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Repair Protocol

For each real mistake:

1. Reproduce the failure on the smallest example, trace, proof, query, command, or design sketch.
2. Name the hidden assumption.
3. Repair the artifact.
4. Save evidence that changed: failing then passing test, corrected proof step, revised diagram, safer command, benchmark, or review note.
5. Add one retrieval card beginning with Check... before... or Do not use... when....

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Mistake Log

Date	Mistake	Symptom	Root cause	Repair evidence	Retrieval card
Starter	Pick one radar row above	Explain how it would fail in this module	Name the assumption	Add a counterexample or corrected artifact	Write the card before closing the page

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Completion Standard

• At least five real mistakes are logged.
• At least two mistakes include a counterexample or failing test.
• At least one mistake connects to an older semester skill.
• At least one correction changes code, a proof, a diagram, a command transcript, a query, or a design decision.