[Mathematics#2] Row Reduction and Echelon Forms (Linear Equations in Linear Algebra#2)

A nonzero row or column

A leading entry of row

: the leftmost nonzero entry

※ entry: values in matrix

Echelon Forms

1. All nonzero rows are above any row 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

Echelon form

Reduced echelon forms

1. The leading entry in each nonzero row is 1.
2. Each leading 1 is the only non zero entry in its column.

Theorem 1. Uniqueness of the Reduced Echelon Forms

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

→ matrix를 row reduction을 통해 reduced echelon form을 만들게 되면, 해당 reduced echelon form의 형태를 제외한 다른 reduced echelon form은 나올 수 없다.

 

Row Reduction Algorithm

step 1) begin with the leftmost nonzero column.

step 2) select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot postion.

step 3) row replacement to create zeros in all positions below the pivot.

step 4) apply steps  1-3 to the submatrix that remain.

※ The combination of steps 1-4 is called fowrad phase → echelon form

 

step 5) beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation. 

※ Step 5 is called backward phase → reduced echelon form

 

solution of linear systems

Ex 1. Find the general solution of the following augmented matrix.

$$ \left[ \begin{matrix} 1 & 6 & 2 & -5 & -2 & 4 \\ 0 & 0 & 2 & -8 & -1 & 3 \\ 0 & 0 & 0 & 0 & 1 & 7 \\ \end{matrix} \right] $$

 

Sol)

Theorem 2. Exsistence and Uniquness Theorem

A linear system is consistent if and only if the rightmost column of hte augmented matrix is not a pivot column- that is, if and only if an echelon form of the augmented matrix has no row of the forms

$$ \left[ \begin{matrix} 0 & 0 & ... & b \end{matrix} \right] $$

$$b\;is \; nonzero$$

 

If a linear system is consistent, then the solution set contains either 

1. a unique solution, when there are no free variables
2. or infinitely many solutions, when there is at least one free variables

 

 

 

 

 

 

 

 

 

※ Reference

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