1.2 - Row Reduction and Echelon Form

Backward phase

The steps that get the matrix into reduced row echelon form (going from bottom right up)

Forward phase

The steps to get it into echelon form (kind of going from the top left down)

A rectangular matrix is in echelon form if it has the following three 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 in the row above it. 3. All entries in a column below a leading entry are zeros (Ex on pg 13)

A matrix is in reduced row echelon form if...

1. All of the conditions for echelon form are true 2. Each leading entry in each nonzero row is 1 3. Each leading 1 is the only nonzero entry in its column (Ex on pg 13)

Using row reduction to solve a linear system

1. Write the augmented matrix of the system 2. Get the matrix into echelon form. If it's inconsistent stop, if not keep going 3. Continue row reduction to get it into reduced row echelon form 4. Write the system of equations corresponding to the matrix obtained in step 3 5. Rewrite each nonzero equation from step 4 so that it's one basic variable is expressed in terms of any free variables appearing in the equation

Pivot column

A column of a matrix that contains a pivot position

Existence and Uniqueness theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is not 0. (Basically saying there can't be a row of all zeros and then a number in the augmented spot) When it is consistent, there's either one solution or infinitely many.

Pivot position

A location in a matrix that corresponds to a leading 1 in the reduced echelon form of the matrix

Uniqueness theorem of the reduced echelon form

Each matrix is row equivalent to one and only one reduced echelon matrix

Basic variable

Variables corresponding to pivot columns

Free variable

Variables that are not a part of a pivot column. Can be any value, which will yield a different set of solutions depending on the value