1. Row Reduction and Echelon Forms (9/9/05) A matrix is in echelon form if: All nonzero rows are above any all-zero rows. Each leading entry of a row is in a column to the right of the leading entry of the row above it. It is in reduced echelon form if: Each leading entry is a 1. Each leading 1 is the only nonzero entry in its column. Theorem: This last form is unique.

2. Pivot Positions and Columns A pivot position is the location of a leading 1 in the reduced echelon form. A pivot column is a column containing a pivot position. A pivot is a nonzero number in a pivot position. A basic variable is one associated with a pivot column. Otherwise, it is called a free variable.

3. The Row Reduction Algorithm For details, see pages 17-20 of our text. The basic idea is: First, moving from upper left to lower right, find a pivot position and clear out below it. This is called the forward phase. Second, moving now back from right to left, clear out above the pivots and change them to 1’s. This is called the backward phase. The result is the unique reduced echelon form for your given matrix.

4. Existence and Uniqueness Theorem A linear system is consistent if and only if an echelon form of the system contains no row of the form [0, 0, …., 0, b] where b is nonzero. If the system is consistent, then it has either a unique solution when there are no free variables, or infinitely many solutions when there is at least one free variable.

5. Solving a Linear System Write the augmented matrix. Use row reduction to create an echelon form and check for consistency. If inconsistent, stop; there are no solutions. Continue, creating the reduced form. Reintroduce the variables in equations. Solve each equation for a basic variable expressed in terms of the free variables.

6. Assignment for Monday Read Section 1.2. Do the Practice Problems and do Exercises 1-15 odd, 21 and 22. We will meet in the classroom Monday (no lab).