Lab 2.1

Gaussian Elimination

Lab goal

Explain why the rows of b must follow the rows of A, then read the solution from three simple triangular solves.

Lecture-note concepts

Code extensionThe full notebook repeats this flow on a seeded 10×10 system and writes out forward and backward substitution.

Key ideas

Row permutationQ A = L U_raw; rhs = Q b: Pivoting reorders the equations, so the same row order must be applied to b.
Diagonal scalingU_raw = D U_unit: The pivots of U_raw become a diagonal matrix; the remaining upper factor has ones on its diagonal.
Triangular solveL y = Qb → D z = y → U_unit x = z: Solve from left to right through L, D, and U_unit instead of forming an inverse.

Lab 2.1 · Guided example

This is a smaller teaching example, not a copy of every notebook cell. It keeps the same factor-and-solve idea with values that fit on one screen.

Select a line to see its action, array shape, and concrete operation.

Simplified fromCh2-1 Gaussian Elimination.ipynb ↗The source uses a random 10×10 system; this trace uses a deterministic 3×3 system.

L01 / 14Create a small system

line action

import numpy as np

All traced lines

Predict

A[0,0] is zero, so another row must supply the first pivot.

Before

Find the vector x that makes A @ x equal b.

A x = b

After

Reveal to compare.

TARGET

A.shapenot stored ?Three equations.

bnot stored ?Three target values.

Continue in Python

Run the original code and change the inputs.