Linear Algebra Workflow: Solve, Analyze, Fit

Problem statement and target quantities

Given a linear system and data-fit branch, solve for state variables, verify numerical consistency, and evaluate conditioning risk before interpreting fitted parameters.

Knowns: matrix A, vector b, regression rows. Unknowns: solution vector, residual diagnostics, and robustness under near-singularity.

Assumptions and unit discipline

Baseline solve assumes square, full-rank A so inverse-based path is defined.
Residual checks are numerical diagnostics; low residual does not guarantee conditioning safety.
Least-squares rows assume linear model structure in parameters.
Maintain consistent variable scaling/units to avoid numeric instability masking model errors.

Derivation checkpoints

A*x = b
x = A^{-1}b  (when A is invertible)
r = A*x - b
beta = (M^T M)^{-1} M^T y

Use residual and determinant/variant rows to separate algebraic solvability from practical numerical reliability.

1. Validate system behavior first

Start by checking whether the system is solvable and stable enough to trust. Compute detA, then solve for x, and immediately inspect residual rows. This turns matrix solving into a verification workflow instead of a blind button press.

2. Inspect structure with eigenvalues and vectors

Use eigenvalues, dot products, and cross products to interpret structure: modes, alignment, and directional sensitivity. These rows help answer “what does this system do?” rather than only “what number did I get?”

3. Move directly into fitting

Build the normal-equation branch in the same sheet so your model assumptions remain explicit. Solving beta0 and beta1 next to matrix checks helps students connect linear algebra to practical estimation tasks.

4. Keep models auditable

Store each scenario as a separate sheet revision (baseline, noisy inputs, near-singular variant). That preserves traceability and makes numerical fragility visible to reviewers instead of hidden in overwritten cells.

5. Document model assumptions and units

Record whether the matrix is expected to be full-rank and note any unit-bearing variables. x=Ainv*b requires an invertible A, and near-singular systems should be treated as conditioning warnings, not trustworthy solutions.

Variable glossary and interpretation

Symbol	Meaning	Interpretation
`A`	Coefficient matrix.	Model structure / coupling.
`b`	Right-hand-side vector.	Known forcing/targets.
`x`	Solved state vector.	Unknown parameters/states.
`r=A*x-b`	Residual vector.	Equation mismatch after solve.
`beta0,beta1`	Linear-fit coefficients.	Intercept and slope in regression branch.

How to interpret the results

Low residual_energy means the solved state satisfies the system numerically.
Compare detA and detA_variant to see how close you are to conditioning failure.
Treat very large invalidity_proxy as a warning to switch to QR/SVD workflows.

Common mistakes and how to avoid them

Using explicit inverse (inv(A)) without monitoring conditioning warnings.
Assuming tiny residual always means a stable model; near-singular systems can still be fragile.
Skipping scaling/normalization when matrix entries differ by many orders of magnitude.

Worked example (interpreted)

Step 1: For the baseline matrix, detA=43 and the solution is x=[1.9302, 2.0233, 1.3721].

Step 2: Residual rows give r=A*x-b and residual_energy=0, confirming a consistent solve.

Step 3: Regression rows yield beta0=0.9 and beta1=0.9.

Interpretation: baseline case is well-behaved; compare against the near-singular variant before trusting sensitivity-heavy conclusions.

When this model is invalid

Condition	Why invalid	Use instead
`detA` near zero	Inverse-based solve becomes ill-conditioned.	Pivoted decomposition (QR/SVD) and condition-number checks.
Underdetermined/overdetermined systems	`A^-1 b` is not defined for non-square or rank-deficient systems.	Least-squares or minimum-norm formulations.
Large scaling mismatch	Numerical error dominates symbolic-looking outputs.	Scale variables and monitor residual-energy rows.

How to validate your worksheet

residual_energy should remain close to zero.
Compare detA vs detA_variant to see conditioning sensitivity.
invalidity_proxy rising sharply flags near-singular behavior risk.

Domain-specific variants in this template

The near-singular branch (A_variant) intentionally stresses inversion stability so teams can see how tiny coefficient changes affect output credibility.

Practice problems (with answer outlines)

Practice 1: Residual verification

After solving Ax=b, compute r and dot(r,r). Interpret non-zero energy.

Outline: Non-zero residual means the computed x does not satisfy equations exactly; inspect conditioning and arithmetic path.

Practice 2: Conditioning alert

Create a near-singular perturbation of one matrix row and compare solution drift.

Outline: Small coefficient changes with large output swing indicate conditioning risk even when residual stays tiny.

Practice 3: Regression branch

Modify y data and recompute beta0, beta1; explain geometric meaning of slope change.

Outline: Use normal equations and compare fitted line tilt/intercept shift against updated data trend.

References and standards

Gilbert Strang, Introduction to Linear Algebra (core concepts). Course/reference page.
Golub & Van Loan, Matrix Computations (numerical methods). Book page.
NIST Dataplot/linear regression references (practical validation workflows). NIST handbook.