Chapter 1: Linear Equations in Linear Algebra 1.1 Systems of Linear Equations

Linear Equation: System of linear equations: A collection of one or more linear equations involving the same variables

Exercise: Determine if each of the following is a system of linear equations.

Solution Set: The set of all possible solutions to a linear system. Definitions: Solution Set: The set of all possible solutions to a linear system. Two linear systems are called equivalent, if they have the same solution set. Exercise: Are the following systems equivalent?

Matrix Notation Any linear system can be written as a matrix which is a rectangular array with m rows and n columns. The size of a matrix is the number of rows and columns it possesses. We write the size as mxn.

For the linear system: The matrix is called the coefficient matrix and the matrix is called the augmented matrix.

Algorithmic Procedure for Solving a Linear System Basic idea: Replace one system with an equivalent system that is easier to solve. Example: Solve the following system:

/ 2 /(-2) R1*(1/2) R1 R2*(-1/2) R2 R1 R2 *(-2) +) -2*R1+R2 R2 /3 R2*(1/3) R2 2*R2+R1 R1

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

Gauss-Jordan Elimination method: to solve a system of linear equations Elementay Row Operations Augmented matrix Reduced augmented matrix

Practice Elementary Row Operations:

Solutions to Linear Systems -- Existence and Uniqueness Is the system consistent; that is, does at least one solution exist? If a solution exists, is it the only one; that is, is the solution unique?

Example 1: A graph of the following system of equations is shown to the right: Is the system consistent? Is it unique?

Example 2: A graph of the following system of equations is shown to the right: Is the system consistent? Is it unique?

Example 3 A graph of the following system of equations is shown to the right: Is the system consistent? Is it unique?

Example 4 A graph of the following system of equations is shown to the right: Is the system consistent? Is it unique?

Find the solution set of each of the following: No solutions Inconsistent (2,1) Unique Consistent Infinitely many solutions. All (x,y) where

Every system of linear equations has either No solutions, or Exactly one solution, or Infinitely many solutions. Geometrically, what does this say about the solution of a linear system?