Chapter1.1 Systems of Linear Equations

In this post, we’ll study the following

• Linear Equation
• Systems of linear Equation
• Solution Set
• Meaning of consistent/inconsistent
• Matrix notation
• Elimination
• Row Operation
• Equivalent/ Row Equivalent

Linear Equation

The variables x_1, x_2, …, x_n is an equation that can be written in the form

where b and the coefficients are real or complex numbers.

A system of linear equation

A system of linear equations is sometimes referred to by the term linear system. A system of linear equations is collection of one or more linear equations involving the same variables.

Solution set

A solution set is the set of all possible solutions to a linear system.

Consistent/Inconsistent

1. Consistent

If the two equations are in a consistent, then (i) there are infinitely many solutions, and (ii) there is one solution. This can be interpreted in two ways.

(i) Infinitely many solutions

(ii) Only one solutions

2. Inconsistent

Inconsistent means no solution.

This means that a system of linear equations has 3 cases

1. no solution ( inconsistent)

2. exactly one solution (consistent)

3. infinitely many solutions (consistent)

Matrix Notation

Matrix notation is a matrix representation of a linear system.

(1) Coefficient matrix

The coefficient matrix is a matrix of only a, excluding b.

Coefficient Matrix

(2) Augmented matrix

The augmented matrix is the matrix up to and including b.

Elimination

The row operation allows you to perform elimination and find the solution of a linear equation.

Make the coefficient of x_1 in the last row zero.

Make the coefficient of x_2 in the last row zero.

Make the coefficient of x_3 in the last row equal to 1.

Now that we know x_3, we can use substitution to find x_1 and x_2.

x_1 = 1

x_2 = 0

x_3 = -1

The three equations are consistent because they have one solution.

Row Operations — row operation

Three row operations are used in the elimination procedure.

(1) replacement

Adding another row to a row multiplied by a constant.

(2) interchange

Swapping two rows

(3) scaling

multiplying everything in a row by a non-zero constant.

Row equivalent

Two matrices are said to be row equivalent if the row operations transform one matrix into the other. If two linear systems are row equivalent, they have the same set of solutions.