Recap · Reading
What you can now do.
Solve any square system the way a computer does, and explain what happened along the way.
A → U by elimination A = LU
- §2.2 Elimination. The café puzzle: subtract one equation from another to remove an unknown. Multiplier = entry ÷ pivot.
- §2.2 Back substitution. The chain of keys: the last row gives the last unknown, which unlocks the row above.
- §2.2 Breakdown. The relay race: a zero pivot means swap rows if you can, and if you can't, the system is singular.
- §2.3 Elimination matrices. Instruction cards: E₂₁ subtracts ℓ times row 1 from row 2. Stack them to reach U.
- §2.6 A = LU. Meal prep: factor once, then solve each new b with two fast triangular solves. L holds the multipliers.
Next lesson: inverses, transposes and permutations (§2.5, §2.7)
The matrix that undoes A, when it exists and how Gauss-Jordan finds it. Then the transpose, symmetric matrices and the permutation matrices behind row exchanges.
Read along
Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), sections 2.2, 2.3 and 2.6. Lectures 2 and 4 of MIT 18.06 on MIT OpenCourseWare cover the same ground.