Unit 1
Systems of Linear Equations

In this unit, we introduce the idea of a system of linear equations, and present a method, called “Gauss-Jordan elimination” for solving such systems. We also introduce the special case of a homogeneous system of equations.

Objectives

When you have completed this unit you should be able to

1. state, from memory, the definition of a “linear equation” and of the “augmented matrix for a system of linear equations.”
2. state, from memory, the elementary row operations that may be used to solve systems of linear equations, and apply these operations to solve linear systems.
3. define what it means for a matrix to be in “row-echelon form” or in “reduced row-echelon form.”
4. find the row-echelon or reduced row-echelon form of a matrix using the method of Gauss-Jordan elimination, and use this technique to solve systems of linear equations.
5. determine whether a system of linear equations is “consistent” or “inconsistent.”
6. give geometric interpretations of systems of linear equations in two and three dimensions, and sketch the different possible cases that may occur in two or three dimensions.
7. define a “homogeneous system of linear equations.”
8. prove that a homogeneous system of linear equations in which the number of unknowns is greater than the number of equations will always have an infinite number of solutions.