Systems of Linear Equations

Basic Notations

Linear equation

An equation that can be written in the following form

…

A system of linear equations (linear system)

Collection of one or more linear equations involving the same variables

ex)

Solution of a linear system

A list (…) of numbers that makes each equation a true statement when the values … are substituted for … respectively.

Equivalent

Two linear systems are called equivalent if they have the same solution set.

Consistency and inconsistency

A linear system is consistent if it has either one or infinitely many solutions; a system is inconsistent if it has no solution.

Matrix Notations

Coefficient matrix and augmented matrix

Given the system

the matrix

is called the coefficient matrix of the above system, and

is called the augmented matrix of the system.

Solving a Linear System

The basic strategy to solve a linear system is to replace one system with an equivalent system that is easier to solve.

Elementary Row Operations

1. (Replacement) Replace one row by the sum of itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.

Given two matrices, if there is a sequence of elementary row operations that transforms one matrix into the other, they are row equivalent.

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Existence and Uniqueness Questions

It is needed to determine that a particular linear system contains either no solution, one solution, or infinitely many solutions.

If the system has one or infinitely many solutions, the system is consistent. Otherwise, it is inconsistent.