Linear Equations I

is $m$ linear equations. Row operations (swap, scale, add multiple of one row to another) preserve the solution set.

Row reduction goal

Produce a row-echelon (or reduced) form: pivots mark basic variables; free variables parameterize a line/plane of solutions when they exist.

Geometry

Each equation is a hyperplane. Solutions are intersections. Parallel distinct planes → empty; coincident → infinitely many.

Lecture depth

Slow pass: definitions, one derivation, two numeric checks, then the exam failure mode.

Derivation sketch

Write the governing identity. Specialize to a 2-variable or n=2 case so arithmetic stays honest. Check a boundary (p=0, det=0, n=1) before trusting the general formula.

Second worked example

Change one parameter from the first example. Predict the direction of change, then compute. If the prediction fails, fix the story before software.

Exam-style checkpoint

1. State the main formula with hypotheses.
2. Compute a numeric instance in under three minutes.
3. Name the modeling failure mode (dependence, singular matrix, biased sample, wrong divisor, etc.).

Common calculator traps

• Mixing degrees/radians is rare here; mixing rate vs mean, /n vs /(n-1), and population vs sample is common.
• For matrices: shape mismatch on multiply, and using det as if it were dimension.

Augmented matrix workflow

Write $[A|b]$, reduce, then translate back to equations. For \times 2$, you should finish by hand in under two minutes once the arithmetic is clean.

References

• MIT OCW 18.06
• soon add