07_Linear_Systems — what does it mean to solve a system of equations?¶
Level: 101 · for anyone who has solved two equations in two unknowns at school
At school a system of equations is something you solve by a trick: substitute, or add the equations, until one unknown is left. A linear algebra book starts the same place, but first it says exactly what is being looked for. A system is a list of tests, a solution is a list of numbers that passes every test, and solving means replacing the system by a simpler one with exactly the same solutions. This chapter follows that start, in the order of the first section of Jim Hefferon's Linear Algebra.
| # | Lesson | The question it answers |
|---|---|---|
| 1 | Linear equations and their solutions | What do the coefficients, constants, tuples and double subscripts in the first definition actually say? |
The through-line¶
Lesson 1 reads the definition. A linear combination is a recipe of fixed coefficients, a linear equation is a test that a tuple passes or fails, and a solution of a system passes every one of its tests. One equation in two unknowns has a whole line of solutions, and a second equation can cut that down to one point. The name linear comes from the two things a linear combination keeps, sums and multiples, which is the link to maps that keep the laws.
What comes next is on the roadmap: why the three steps of Gauss's method never change the set of solutions, and why the three possible answers are one solution, none, or infinitely many.
Where this is taught¶
The first section of Jim Hefferon, Linear Algebra ↗, free to download with answers to every exercise. Gilbert Strang's Introduction to Linear Algebra covers the same ground in its chapter on solving linear equations, with more pictures and fewer definitions. For which book to read first, and what to know before starting, see Linear algebra: a reading guide.
A note on the code¶
Every number is a fractions.Fraction, so a tuple satisfies an equation when the two sides are equal, not nearly equal. Solving a system in floating point is a different question, how many digits the answer keeps, and it is the one 01_Precision prepares for.
Auf Deutsch: Stichwörter¶
Was es heißt, ein Gleichungssystem zu lösen: eine Gleichung ist ein Test, eine Lösung besteht jeden.
Stichwörter: lineares Gleichungssystem, Lösung, Lösungsmenge, Unbekannte, Gauß-Verfahren, Hefferon.