Chapter 2 · Lesson 2 · Strang §2.1 to §2.4, §2.6

Solving Ax = b

By the end of this lesson you can

Solve a system of equations the way a computer does
Multiply matrices by rows and by columns, and know why the order matters
Factor A = LU once and reuse it to solve fast

Two ways to read Ax = b, then how a computer solves it: subtract rows until the system is a triangle, solve from the bottom up, multiply matrices, and record the steps as A = LU. 8 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.

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Slides

Lesson 2Rows and columnsSolving Ax = b: the row picture and the column pictureOpen slide →01 · Strang §2.1 · Vectors and linear equations

Row Picture vs Column Picture of Ax = b

Two ways to see a linear system: lines crossing, or columns combining to make b. Interactive plot and Python in the browser. Strang §2.1.

Lesson 2EliminationElimination: remove one unknown at a timeOpen slide →02 · Strang §2.2 · The idea of elimination

Gaussian Elimination Method Explained Visually

How elimination solves two equations: subtract to remove an unknown, then back-substitute. A café puzzle, a worked example and live Python. Strang §2.2.

Lesson 2PivotsPivots and back substitution: clear the columns, then solve from the bottom upOpen slide →03 · Strang §2.2 · Pivots

Pivots and Back Substitution: 3×3 Elimination Step by Step

Clear one column at a time with the pivots, reach an upper triangular U, then solve from the bottom up. Worked 3×3 example with Python. Strang §2.2.

Lesson 2Zero pivotsZero pivots: swap rows, or the system is singularOpen slide →04 · Strang §2.2 · Breakdown of elimination

When Elimination Fails: Zero Pivots and Singular Systems

A zero pivot means swap rows or stop. See the three cases in the row picture: one solution, no solution, infinitely many. Strang §2.2.

Lesson 2Elimination matricesElimination matrices: each row step is a matrixOpen slide →05 · Strang §2.3 · Elimination using matrices

Elimination Matrices: Row Operations as Matrix Multiplication

Every elimination step is a matrix E. Multiplying by E subtracts a multiple of one row from another. Build E21 and E31 in Python. Strang §2.3.

Lesson 2Matrix times matrixMultiplying two matricesOpen slide →06 · Strang §2.4 · Rules for matrix operations

Matrix Multiplication Explained: Row Times Column, Step by Step

How to multiply two matrices: each entry is a row times a column (a dot product), and each column of AB is A times a column of B. Worked examples, sizes, why AB is not BA, and Python. Strang §2.4.

Lesson 2A = LULU decomposition: A = LU keeps a record of eliminationOpen slide →07 · Strang §2.6 · Elimination = factorization

LU Decomposition Method Explained: Why A = LU

Elimination turns A into U, and the multipliers fill a lower triangular L so that A = LU. Factor once, solve many times. Python example. Strang §2.6.

Lesson 2RecapWhat you can now do.Open slide →08 · Recap · Reading

Lesson 2 Recap: Elimination and A = LU

Lesson 2 in one page: row and column pictures, multipliers and pivots, back substitution, zero pivots and row exchanges, elimination matrices, matrix multiplication, and A = LU.