**Terminology:**- linear system, solution set, consistent, inconsistent, elementary row operations, unique solution, echelon form, reduced echelon form, pivot position, pivoting, basic variable, free variable, general solution, parametric solution, back-substitution.
**Objectives:**- learn what a linear system is and what kind of solutions are possible; learn how to solve a linear system using elementary row operations and how to describe the set of solutions for the linear system.
**Reading Assignment:**- Chapter 1, Sections 1.1-1.2 (pages 1-26).
**Lesson Outline****Key Ideas and Discussion:**- To start, you need to understand what a linear system is and why a linear
system can have three types of solution: a unique solution, infinitely many
solutions, or no solution. Associated with any linear system is a coefficient
matrix. For ease of solution, the coefficient matrix is usually
augmented with a column of right hand sides for the linear system.
Solving a linear system consists of a series of steps designed to gradually
replace the original system with a sequence of simpler systems. Each
of the simpler systems has a solution set which is the same as the
solution set for the original
system. The steps can be organized so that eventually you are left with a
system which has one equation with only one unknown. This equation is solved
and the information is used successively in other equations until all of the
unknowns are determined. All of the steps for the solution to a linear system
can be carried out through
elementary row operations
on the augmented coefficient matrix.
A systematic method for solving a linear system uses elementary row operations (

*ero*'s) to reduce the augmented matrix to echelon form. The echelon form can be used to determine what type of solutions the system has. When a solution exists, back-substitution can be used with the echelon form to find one. In the case where there is more than one solution, additional ero's can be used to determine the reduced echelon form, and this form can be used to determine a general (parametric) form for all solutions. The general solutions will depend on one or more free variables (free parameters) which can be selected arbitrarily, to provide an infinite set of solutions to the original linear system. **Practice Problems:**- 1.1.3, 5, 11, 15,
19,
23 (pp. 10-11);
1.2.5,
9, 15, 19, 23, 25 (pp. 25-26).
**Assignment**

1999-09-22