System of Linear Equations
Nattee Niparnan
LINEAR EQUATIONS
Linear Equation
• An Equation
– Represent a straight line
– Is a “linear equation” in the variable x and y.
• General form
– ai a real number that is a coefficient of xi
– b  another number called a constant term
System of a Linear Equation
• A collection of several linear equations
– In the same variables
• What about
– A linear equation
• in the variables x1, x2 and x3
– Another equation
• in the variables x1, x2,x3 and x4
– Do they form a system of linear equation?
Solution
• A linear equation
• Has a solution
• When
• It is called a solution to the system if it is a
solution to all equations in the system
Number of Solution
• Solution can have
– No solution
– One solution
– Infinite solutions
Example 1
• Show that
– For any value of s and t
– xi is the solution to the system
Example 1 Solution
Parametric Form
• Solution of the system in Equation 1 is
described in a
– It is given as a function in
– It is called a
s and t
of the system
• Every linear equation system having solutions
– Can be written in parametric form
Try another one
• Solve it using parametric form
• In term of x and z
• In term of y and z
There are
several general
solutions
Geometrical Point of View
• In the case of 2 variables
– Each equation is represent a line in 2D
– Every point in the line satisfies the equation
• If we have 2 equations
– 3 possibilities
• Intersect in a point
• Intersect as a line
• Parallel but not intersect
As a point
No
intersection
As a line
3D Case
• What does
represent?
3D Case
• A plane
Higher Space?
• Somewhat difficult to imagine
– But Linear Algebra will, at least, provides some
characteristic for us
Cogito, ergo sum
I also speak
Calculus
MANIPULATING THE SYSTEM
Augmented Matrix
Augmented
matrix
Coefficient
matrix
Constant
matrix
Equivalent System
• System  a set of linear
equations
– Two systems having the
same solution is said to be
“
”
• Some system is easier to
identify the solution
• To solve a system, we
manipulate it into an
“easy” system that is still
equivalent to the original
system
System 1
Solution
preserve
operation
System 2
Solution
preserve
operation
System 3
Elementary Operation
Solved!
Elementary Operation
• Interchange two equations
• Multiply one equation with a
• Add a multiple of one equation to a
equation
number
Theorem 1
• Suppose that an elementary operation is
performed on a linear equation system
– Then, there solution are still the same
Proof
Elementary Row Operation
• We don’t really do the elementary operation
• We write the system as an augmented matrix
and then perform “
” on that matrix
Goal of Elementary Operation
• To arrive at an easy system
GAUSSIAN ELIMINATION
Gaussian Elimination
• An algorithm that manipulate an augmented
matrix into a “nice” augmented matrix
Row Echelon Form
• A matrix is in “Row Echelon Form” (called row
echelon matrix) if
– All zero rows are at the bottom
– The first nonzero entry from the left in each
nonzero row is 1
• (that 1 is called a leading 1 of that row)
– Each leading 1 is to the right of all leading 1’s in
the row above it
Example
Echelon?
• Diagonal Formation
Reduced Row Echelon
• The leading 1 is the only nonzero element in
that column
row echelon
Reduced row echelon
Theorem 2
• Every matrix can be manipulated into a
(reduced) row echelon form by a series of
elementary row operations
Using (Reduced) Row Echelon Form
Using (Reduced) Row Echelon Form
No solution
Solution to (c)
Variable corresponding
to the leading 1’s is
called “leading
variable”
The non-leading
variables end up as a
parameter in the
solution
Gaussian Elimination
• If the matrix is all zeroes  stop
• Find the first column from the left containing a
non zero entry (called it A) and move the row
having that entry to the top row
• Multiply that row by 1/A to create a leading 1
• Subtract multiples of that row from rows below
it, making entry in that column to become zero
• Repeat the same step from the matrix consists of
remaining row
Gauss?
Redundancy
Subtract 2 time row 1
from row 2
And
Subtract 7 time row 1
from row 3
Subtract 2 time row 2
from row 1
And
Subtract 3 time row 2
from row 3
Redundancy
Observe that the last
row is the triple of the
second row
Back Substitution
• Gaussian Elimination brings the matrix into a
row echelon form
– To create a reduced row echelon form
• We need to change step 4 such that it also create zero
on the “above” row as well
• Usually, that is less efficient
• It is better to start from the row echelon form
and then use the leading 1 of the bottommost row to create zero
Example
Example
Another Example
Try it
Solution
Must be
0
Rank
• It is (later) shown that, for any matrix A, it has the
same “Reduced row echelon form”
– Regardless of the elementary row operation performed
• But it s not true for “row echelon form”
– Different sequence of operations leads to different row
echelon matrix
• However, the number of leading 1’s is always the same
– Will be proved later
• Hence, the number of leading 1’s depends on A
Theorem 3
• Suppose a system of equation on
variables has a solution, if the rank of the
augmented matrix is
– the set of the solution involve exactly
parameters
Homogeneous Equation
When b = 0
What is the
solution?
Homogeneous Linear System
• Xi = 0 is always a solution to the homogeneous
system
– It is called “trivial” solution
• Any solution having nonzero term is called
“nontrivial” solution
Existence of Nontrivial Solution to the
homogeneous system
• If it has non-leading entry in the row echelon
form
– The solution can be described as a parameter
• Then it has nonzero solution!!!
– Nontrivial
• When will we have non-leading entry?
– When we have more variable than equation
GEOMETRICAL VIEW OF LINEAR
EQUATION
Geometrical Point of View
• A system of Linear Equation
Column Vector view
Network Flow Problem
• A graph of traffic
– Node = intersection
– Edge = road
– Do we know the flow at each road?
Network Flow Problem
• Rules
– For each node, traffic in equals traffic out
Formulate the System
• Five equations, six vars
Solve it