Section 2 Definitions

Echelon Forms

  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
  3. All entries in a column below a leading entry are zero

REDUCED ECHELON
The above three plus

  1. Leading entry in each row is 1
  2. 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.

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