Systems of Linear Equations
and Row Echelon Form
Motivation
• Physical systems typically involve many
different quantities.
• Relationships between quantities give rise to a
system of equations.
• Nonlinear equations can be approximated by
linear equations.
• Linear Algebra is the study of linear systems
and efficient methods for solving them.
Linear Equations
Linear Equations
Nonlinear Equations
Linear Equations
System of Linear Equations
Solution Set
Solution Set
Consistent Linear Systems
A linear system is consistent if it has at least
one solution.
A linear system is inconsistent if it has no
solutions.
Two Equations, Two Unknowns:
Lines in a Plane
Three Possible Types of Solutions
A unique solution
Three Possible Types of Solutions
No solution
Three Possible Types of Solutions
Infinitely many solutions
Three Equations, Three Unknowns:
Planes in Space
Intesections of Planes
What type of solution sets are represented?
A System of Linear Equations
Augmented Matrix
Coefficient Matrix
Example
Write as an augmented matrix:
Example
Write as a set of linear equations:
Solve the System
Solve the System
Elementary Operations
• Multiply any equation by a nonzero
number.
• Replace any equation with itself
added to a multiple of another
equation.
• Interchange the order in which the
equations are listed.
Row Operations
• Multiply any row by a nonzero
number.
• Replace any row by a multiple of
another row added to it.
• Switch two rows.
What is the “nicest” form
of a reduced matrix?
• What happens if the coefficient
matrix is reduced to the identity
matrix?
• Can the coefficient matrix always be
reduced to the identity matrix?
Reduced Row Echelon Form
A rectangular matrix is in reduced row echelon form
if it has the following conditions:
1. If a row has nonzero entries, then the first
nonzero entry is a 1, called the leading 1 (or pivot) in
this row
2. If a column has a leading 1, the all the other
entries in that column are 0.
3. If a row contains a leading 1, then each row
above it contains a leading 1 further to the left.
Note: Condition 3 implies that rows of 0’s, if any,
appear at the bottom of the matrix.
Reduced Row Echelon Form?
Reduced Row Echelon Form?
Pivot Positions and Pivot Columns
Suppose row operations are used to transform
matrix to Reduced Row Echelon form. Then:
1. The positions of the first nonzero entry in
each row are called the pivot positions.
2. The columns containing a pivot position are
called the pivot columns.
What are the pivot positions
and pivot columns?
Types of Variables
The variables corresponding to the columns of
a matrix that are not pivot columns are called
the free variables. These variables are
assigned parameters. The other variables are
called basic variables or lead variables and
may be solved in terms of the parameters.
Solve the System
Solve the System
Solve the System
Types of Solutions
Suppose a linear system [A|b] is given where A has m rows
and n columns:
1. The system is inconsistent if the augmented column is a
pivot column.
2. The system is consistent if the augmented column is not a
pivot column.
a. There is a unique solution if
The number of pivot columns = n
b. There are an infinite number of solutions if
The number of pivot columns < n