Chapter 1 — Systems of Linear Equations
Many unknowns, many conditions, every condition a sum of number × unknown — how to tell whether such a system has a solution, and how to find all of them.
A linear equation is the humblest kind of equation there is: each unknown appears on its own, multiplied by a number, and the results are added up. One such equation in two unknowns is just a straight line. Yet put many of them together and they describe a remarkable range of the world. An engineer checking a steel roof truss writes two balance equations at every joint, one for horizontal and one for vertical forces, and gets dozens of equations in dozens of unknown member forces. A chemist balancing a reaction requires the number of atoms of each element to match on both sides. A traffic planner insists that the cars entering every junction equal the cars leaving it. Wassily Leontief won the 1973 Nobel Prize in Economics for describing a whole national economy — how much each industry must produce for all the others — as one large linear system.
In every case the questions are the same. Is there a solution at all? Is there only one, or many? And how do we find them, quickly and reliably, when there are far too many equations to juggle in our heads? This chapter answers all three. The answer is an algorithm — elimination — that you will use in every later chapter of the course, and that computers run on systems with millions of unknowns.
- recognise linear equations, check whether a list of numbers is a solution, and describe a solution set with parameters;
- explain geometrically, with lines and planes, why a linear system has no solution, exactly one, or infinitely many;
- write a system as an augmented matrix and simplify it with the three elementary row operations, knowing why they never change the solutions;
- reduce any matrix to row echelon and reduced row echelon form by Gaussian and Gauss–Jordan elimination, and read off the solution set, including free variables;
- set up and solve linear systems from applications: curve fitting, networks and flows, electrical circuits and chemical equations.
Sections in this chapter
Introduction to Linear Systems
Linear equations, solution sets, the geometry of lines and planes, augmented matrices, row operations and back-substitution.
Gaussian and Gauss–Jordan Elimination
Row echelon and reduced row echelon form, pivots and free variables, existence and uniqueness, homogeneous systems — the systematic algorithm for every system.
Applications of Linear Systems
Flows in networks, electrical circuits, balancing chemical equations, fitting polynomials through data, mixtures and partial fractions.