14Echelon FormA rectangular matrix is in echelon form if it has the following properties:1. All nonzero rows are above any rows of all zeros.2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
17Pivot Positions and Pivot Columns The positions of the first nonzero entry in each row are called the pivot positions.The columns containing a pivot position are called the pivot columns.
18Types of Solutions1. No solution – the augmented column is a pivot column.2. A unique solution – every column except the augmented column is a pivot column.3. An infinite number of solutions – some column of the coefficient matrix is not a pivot column.The variables corresponding to the columns that are not pivot columns are assigned parameters. These variables are called the free variables. The other variables may be solved in terms of the parameters and are called basic variables or leading variables.
24Echelon FormA rectangular matrix is in row reduced echelon form if it has the following properties:1. It is in echelon form.2. All entries in a column above and below a leading entry are zero.3. Each leading entry is a 1, the only nonzero entry in its column.
29ExampleEstimate the temperatures T1, T2, T3, T4, T5, and T6 at the six points on the steel plate below. The value Tk is approximated by the average value of the temperature at the four closest points.202020T1T2T3T4T5T6202020
30Number of free variable = n – rank(A) The rank of a matrix is the number of nonzero rows in its row echelon form.Rank TheoremLet A be the coefficient matrix of a system of linear equations with n variables. If the system is consistent, thenNumber of free variable = n – rank(A)