Section 2 Definitions

#### Echelon Forms

- All nonzero rows are above any rows of all zeros
- Each leading entry of a row is in a column to the right of the leading entry of the row above it
- All entries in a column
**below**a leading entry are zero

**REDUCED ECHELON**

The above three plus

- Leading entry in each row is 1
- Each leading 1 is the only non-zero entry in its column

#### Row Reductions

(1)\begin{align} \begin{bmatrix} 3 & 4 & 1 & 2 & | & 2 \\ 0 & 2 & 3 & -1 & | & 0\\ 0 & 0 & 4 & 6 & | & 8 \\ 0 & 0 & 0 & 2 & | & 4 \\ \end{bmatrix} = \begin{bmatrix} 1&0&0&0&3 \\ 0&1&0&0&6 \\ 0&0&1&0&9 \\ 0&0&0&1&7 \\ \end{bmatrix} \text{Left matrix is row reduced into the right which is in proper Reduced Echelon Form} \end{align}

#### Pivot Positions

A **pivot postition** in a matrix A is a location in A that corresponds to a leading entry 1 in the reduced echelon form of A.

A **pivot column** is a column of A that contains a pivot position.

The variables that correspond to pivot columns in a matrix are called **basic variables**.

A **free variable** is a variable that is not a basic variable. A free variable can have any value and ensures that there is infinitely many solutions in a consistent system.

A **unique solution** has no free variables and a pivot in each column.

page revision: 7, last edited: 10 Feb 2015 00:12