Euclidean Euclidean mm-Space & Linear Equations-Space & Linear Equations

Row Reduction of Row Reduction of Linear SystemsLinear Systems

Matrix

A matrix is a rectangular array of numbers.This array is enclosed in brackets or parentheses.Each number in a matrix is called and element or

entry of that matrix.Each horizontal line of numbers in a matrix is

called a row.Each vertical line of numbers in a matrix is called

a column.

Augmented Matrix

Matrix arise from a linear system in standard form by deleting all unknowns operations and equal signs.

A vertical line is drawn along the equal signs to separate the constant column.

When an unknown is missing in an equation, the coefficient zero is inserted in the augmented matrix.

Elementary Row Operations

Multiply a row by a nonzero scalar.

Interchange the positions of two rows.

Replace a row by the sum of itself and a multiple of another row.

Example

x1 + x2 + 2x3 = 8

-x1 - 2x2 + 3x3 = 1

3x1 - 7x2 + 4x3 = 10

Row-Equivalent Matrices

When one matrix can be obtained from another matrix by a finite sequence of row operations, the two matrices are said to be row-equivalent.

The underlying linear systems of two row-equivalent matrices have same solutions.

Row-Reduced Echelon Form

A matrix is in row reduced echelon form if it satisfies the following conditions:

1. The first nonzero entry in each row is a 1, the leading 1.

2. All other entries of a column containing a leading 1 are zeros.

3. In any two rows with some nonzero entries, the leading one of the top row is farther to the left.

4. Rows with all zeros are placed below rows with some nonzero entries.

Gauss-Jordan Elimination

The process of transforming an augmented matrix to an equivalent row-reduced echelon form is called the Gauss-Jordan Elimination.

Row Echelon Form

A matrix is in row echelon form if it satisfies the following conditions:

1. The leading nonzero entry in each row is a 1.

2. All elements below a leading 1 are zeros.

3. In any two rows with leading 1s, the leading 1 of the top row is farther to the left.

4. Rows with all zeros are placed below rows with some nonzero entries.

Homework 2.3