Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Math 407A: Linear Optimization
Lecture 2
Math Dept, University of Washington
January 7, 2010
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
Gauss-Jordan Elimination (Pivoting)
What is linear programming?
Applications of Linear Programing
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
The set of solutions is either
.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
The set of solutions is either empty,
.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
The set of solutions is either empty, a single point,
.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
The set of solutions is either empty, a single point, or an infinite
set.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Solving Systems of Linear equations
Let A ∈ Rm×n and b ∈ Rm .
Find all solutions x ∈ Rn to the system Ax = b.
The set of solutions is either empty, a single point, or an infinite
set.
If a solution x0 ∈ Rn exists, then the set of solutions is given by
x0 + Nul (A) .
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
The three elementary row operations.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
The three elementary row operations.
1. Interchange any two rows.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
The three elementary row operations.
1. Interchange any two rows.
2. Multiply any row by a non-zero constant.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
The three elementary row operations.
1. Interchange any two rows.
2. Multiply any row by a non-zero constant.
3. Replace any row by itself plus a multiple of any other row.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination and the 3 Elementary Row
Operations
We solve the system Ax = b by transforming the augmented matrix
[ A | b]
into upper echelon form using the three elementary row operations.
This process is called Gaussian elimination.
The three elementary row operations.
1. Interchange any two rows.
2. Multiply any row by a non-zero constant.
3. Replace any row by itself plus a multiple of any other row.
These elementary row operations can be interpreted as multiplying
the augmented matrix on the left by a special matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Exchange and Permutation Matrices
An exchange matrix is given by permuting any two columns of the
identity.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Exchange and Permutation Matrices
An exchange matrix is given by permuting any two columns of the
identity.
Multiplying any 4 × n matrix on the left by the exchange matrix


1 0 0 0
 0 0 0 1 


 0 0 1 0 
0 1 0 0
will exchange the second and fourth rows of the matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Exchange and Permutation Matrices
An exchange matrix is given by permuting any two columns of the
identity.
Multiplying any 4 × n matrix on the left by the exchange matrix


1 0 0 0
 0 0 0 1 


 0 0 1 0 
0 1 0 0
will exchange the second and fourth rows of the matrix.
(multiplication of a m × 4 matrix on the right by this exchanges
the second and fourth columns.)
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Exchange and Permutation Matrices
An exchange matrix is given by permuting any two columns of the
identity.
Multiplying any 4 × n matrix on the left by the exchange matrix


1 0 0 0
 0 0 0 1 


 0 0 1 0 
0 1 0 0
will exchange the second and fourth rows of the matrix.
(multiplication of a m × 4 matrix on the right by this exchanges
the second and fourth columns.)
A permutation matrix is obtained by permuting the columns of the
identity matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Left Multiplication of A:
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Left Multiplication of A:
When multiplying A on the left by an m × m matrix M, it is often
useful to think of this as an action on the rows of A.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Left Multiplication of A:
When multiplying A on the left by an m × m matrix M, it is often
useful to think of this as an action on the rows of A.
For example, left multiplication by a permutation matrix permutes
the rows of the matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Left Multiplication of A:
When multiplying A on the left by an m × m matrix M, it is often
useful to think of this as an action on the rows of A.
For example, left multiplication by a permutation matrix permutes
the rows of the matrix.
However, mechanically, left multiplication corresponds to matrix
vector multiplication on the columns.
MA = [Ma•1 Ma•2 · · · Ma•n ]
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Right Multiplication of A:
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Right Multiplication of A:
When multiplying A on the right by an n × n matrix N, it is often
useful to think of this as an action on the columns of A.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Right Multiplication of A:
When multiplying A on the right by an n × n matrix N, it is often
useful to think of this as an action on the columns of A.
For example, right multiplication by a permutation matrix
permutes the columns of the matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Right Multiplication of A:
When multiplying A on the right by an n × n matrix N, it is often
useful to think of this as an action on the columns of A.
For example, right multiplication by a permutation matrix
permutes the columns of the matrix.
However, mechanically, right
multiplication corresponds to
left matrix vector multiplication
on the rows.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Notes on Matrix Multiplication
Let A = [aij ]m×n ∈ Rm×n .
Right Multiplication of A:
When multiplying A on the right by an n × n matrix N, it is often
useful to think of this as an action on the columns of A.
For example, right multiplication by a permutation matrix
permutes the columns of the matrix.

a1• N
However, mechanically, right
 a2• N

multiplication corresponds to
AN =  .
 ..
left matrix vector multiplication
am• N
on the rows.
Lecture 2
Math 407A: Linear Optimization





Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
The key step in Gaussian elimination is to transform a vector of
the form


a
 α ,
b
where a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1 , into one of the form


a
 α  .
0
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
The key step in Gaussian elimination is to transform a vector of
the form


a
 α ,
b
where a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1 , into one of the form


a
 α  .
0
This can be accomplished by left matrix multiplication as follows.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1

Ik×k
 0
0


0
0
a
 α 
1
0
−1
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1

Ik×k
 0
0

 
0
0
a
 α  = 
1
0
−1
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization

.
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1

Ik×k
 0
0


 
0
0
a
a
.
 α  = 
1
0
−1
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1

Ik×k
 0
0


 
0
0
a
a
 α  =  α .
1
0
−1
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
a ∈ Rk , 0 6= α ∈ R, and b ∈ Rn−k−1

Ik×k
 0
0


 
0
0
a
a
 α  =  α .
1
0
−1
−α b I(n−k−1)×(n−k−1)
0
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
The matrix

Ik×k
 0
0

0
0

1
0
−1
−α b I(n−k−1)×(n−k−1)
is called a Gaussian elimination matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
The matrix

Ik×k
 0
0

0
0

1
0
−1
−α b I(n−k−1)×(n−k−1)
is called a Gaussian elimination matrix.
This matrix is invertible with inverse


Ik×k
0
0
 0
.
1
0
−1
0
α b I(n−k−1)×(n−k−1)
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination Matrices
The matrix

Ik×k
 0
0

0
0

1
0
−1
−α b I(n−k−1)×(n−k−1)
is called a Gaussian elimination matrix.
This matrix is invertible with inverse


Ik×k
0
0
 0
.
1
0
−1
0
α b I(n−k−1)×(n−k−1)
Note that a Gaussian elimination matrix and its inverse are both
lower triangular matrices.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Sub-Algebras
Lower (upper) triangular matrices in Rn×n are said to form a
sub-algebra of Rn×n .
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Sub-Algebras
Lower (upper) triangular matrices in Rn×n are said to form a
sub-algebra of Rn×n .
A subset S of Rn×n is said to be a sub-algebra of Rn×n if
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Sub-Algebras
Lower (upper) triangular matrices in Rn×n are said to form a
sub-algebra of Rn×n .
A subset S of Rn×n is said to be a sub-algebra of Rn×n if
I
S is a subspace of Rn×n ,
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Sub-Algebras
Lower (upper) triangular matrices in Rn×n are said to form a
sub-algebra of Rn×n .
A subset S of Rn×n is said to be a sub-algebra of Rn×n if
I
S is a subspace of Rn×n ,
I
S is closed wrt matrix multiplication, and
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Sub-Algebras
Lower (upper) triangular matrices in Rn×n are said to form a
sub-algebra of Rn×n .
A subset S of Rn×n is said to be a sub-algebra of Rn×n if
I
S is a subspace of Rn×n ,
I
S is closed wrt matrix multiplication, and
I
if M ∈ S is invertible, then M −1 ∈ S.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Transformation to echelon
(upper triangular) form.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Lecture 2
Math 407A: Linear Optimization

1 1 2
A =  2 4 2 .
−1 1 3
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.

1 1 2
A =  2 4 2 .
−1 1 3
Eliminate the first column with
a Gaussian elimination matrix.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.
Lecture 2
Math 407A: Linear Optimization

1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.

1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1



1 0 0
1 1 2
G1 A =  −2 1 0   2 4 2 
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 0 0
1 1 2
G1 A =  −2 1 0   2 4 2  = 
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
=
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
=
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
=
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization
2


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
= 0
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization
2


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
= 0 2
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization
2


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
= 0 2
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization

2
−2 
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice

Transformation to echelon
(upper triangular) form.
Eliminate the first column with
a Gaussian elimination matrix.


1 1 2
A =  2 4 2 .
−1 1 3


1 0 0
G1 =  −2 1 0 
1 0 1

 
1 1
1 0 0
1 1 2





2 4 2
G1 A = −2 1 0
= 0 2
0 2
1 0 1
−1 1 3
Lecture 2
Math 407A: Linear Optimization

2
−2 
5
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0
Lecture 2
Math 407A: Linear Optimization
eliminiation on the second column.

1
2
2 −2 
2
5
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0
Lecture 2
Math 407A: Linear Optimization
eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1
0 0
1 1
2
 0
1 0   0 2 −2  = 
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1
1
0 0
1 1
2
 0
1 0   0 2 −2  = 
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0
0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1 1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0
0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1 1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0
0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization
2


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1 1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0 2
0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization
2


Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1 1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0 2
0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization

2
−2 
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gaussian Elimination in Practice
Now do Gaussian

1
 0
0

eliminiation on the second column.



1
0 0
1
2
1 0 
2 −2 
G2 =  0
0 −1 1
2
5



1 1
1
0 0
1 1
2
 0
1 0   0 2 −2  =  0 2
0 0
0 −1 1
0 2
5
Lecture 2
Math 407A: Linear Optimization

2
−2 
7
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?



Ik×k −α−1 a
0
a
 0
 α 
α−1
0
−1
0
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

 

Ik×k −α−1 a
0
a
 0
 α  = 
α−1
0
−1
0
−α b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization

.
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

  

Ik×k −α−1 a
0
0
a
 0
 α  =  .
α−1
0
0
−α−1 b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

  

Ik×k −α−1 a
0
0
a
 0
 α  =  1 .
α−1
0
0
−α−1 b I(n−k−1)×(n−k−1)
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

  

Ik×k −α−1 a
0
0
a
 0
 α  =  1 .
α−1
0
0
−α−1 b I(n−k−1)×(n−k−1)
0
b
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

  

Ik×k −α−1 a
0
0
a
 0
 α  =  1 .
α−1
0
0
−α−1 b I(n−k−1)×(n−k−1)
0
b
What is the inverse of this matrix?
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Gauss-Jordan Elimination, or Pivot Matrices
What happens in the following multiplication?

  

Ik×k −α−1 a
0
0
a
 0
 α  =  1 .
α−1
0
0
−α−1 b I(n−k−1)×(n−k−1)
0
b
What is the inverse of this matrix?


Ik×k a
0
.
 0
α
0
0
b I(n−k−1)×(n−k−1)
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Math 407 Begins!
MATH 407 NOW BEGINS!!
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is optimization?
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is optimization?
A mathematical optimization problem is one in which some
function is either maximized or minimized relative to a given set of
alternatives.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is optimization?
A mathematical optimization problem is one in which some
function is either maximized or minimized relative to a given set of
alternatives.
I
The function to be minimized or maximized is called the
objective function.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is optimization?
A mathematical optimization problem is one in which some
function is either maximized or minimized relative to a given set of
alternatives.
I
The function to be minimized or maximized is called the
objective function.
I
The set of alternatives is called the feasible region (or
constraint region).
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is optimization?
A mathematical optimization problem is one in which some
function is either maximized or minimized relative to a given set of
alternatives.
I
The function to be minimized or maximized is called the
objective function.
I
The set of alternatives is called the feasible region (or
constraint region).
I
In this course, the feasible region is always taken to be a
subset of Rn (real n-dimensional space) and the objective
function is a function from Rn to R.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem in finitely many
variables having a linear objective function and a constraint
region determined by a finite number of linear equality and/or
inequality constraints.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem
in finitely many variables
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem
in finitely many variables
having a linear objective function
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem
in finitely many variables
having a linear objective function
and a constraint region determined by a
finite number of constraints
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
What is linear programming (LP)?
A linear program is an optimization problem
in finitely many variables
having a linear objective function
and a constraint region determined by a
finite number of constraints
that are linear equality and/or linear inequality constraints.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
I
A linear function of the variables x1 , x2 , . . . , xn is any function
of the form
c 1 x1 + c 2 x2 + · · · + c n xn
for fixed ci ∈ R i = 1, . . . , n.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
I
I
A linear function of the variables x1 , x2 , . . . , xn is any function
of the form
c 1 x1 + c 2 x2 + · · · + c n xn
for fixed ci ∈ R i = 1, . . . , n.
A linear equality constraint is any equation of the form
a1 x1 + a2 x2 + · · · + an xn = α,
where α, a1 , a2 , . . . , an ∈ R.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
I
I
A linear function of the variables x1 , x2 , . . . , xn is any function
of the form
c 1 x1 + c 2 x2 + · · · + c n xn
for fixed ci ∈ R i = 1, . . . , n.
A linear equality constraint is any equation of the form
a1 x1 + a2 x2 + · · · + an xn = α,
I
where α, a1 , a2 , . . . , an ∈ R.
A linear equality constraint is any inequality of the form
a1 x1 + a2 x2 + · · · + an xn ≤ α,
or
a1 x1 + a2 x2 + · · · + an xn ≥ α,
where α, a1 , a2 , . . . , an ∈ R.
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Compact Representation
minimize
c 1 x1 + c 2 x2 + · · · + c n xn
subject to ai1 xi + ai2 x2 + · · · + ain xn ≤ αi
bi1 xi + bi2 x2 + · · · + bin xn = βi
Lecture 2
Math 407A: Linear Optimization
i = 1, . . . , s
i = 1, . . . , r .
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Vector Inequalities: Componentwise
Let x, y ∈ Rn .



x =

x1
x2
..
.
xn








y =

y1
y2
..
.





yn
We write x ≤ y if and only if
xi ≤ yi , i = 1, 2, . . . , n .
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Notation
c 1 x1 + c 2 x2 + · · · + c n xn = c T x



c =

c1
c2
..
.
cn
Lecture 2
Math 407A: Linear Optimization








x =

x1
x2
..
.





xn
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Notation
ai1 xi + ai2 x2 + · · · + ain xn ≤ αi i = 1, . . . , s
⇐⇒
Ax ≤ a



A=

Lecture 2
Math 407A: Linear Optimization

a11 a12 . . . a1n
a21 a22 . . . a2n 

..
.. . .
.. 
. . 
.
.
as1 as2 . . . asn



a=

α1
α2
..
.





αn
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Matrix Notation
bi1 xi + bi2 x2 + · · · + bin xn = βi
i = 1, . . . , r
⇐⇒
Bx = b



B=

Lecture 2
Math 407A: Linear Optimization

b11 b12 . . . b1n
b21 b22 . . . b2n 

..
.. . .
.. 
.
.
.
. 
br 1 br 2 . . . brn



b=

β1
β2
..
.





βs
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
LP’s Matrix Notation
minimize c T x
subject to Ax ≤ a and Bx = b





c1
α1
β1
 .. 
 .. 
 ..
c =  . , a =  . , b =  .
cn
αn
βs



a11 . . . a1n
b11 . . .



.
..
..
A=
, B = 
.
as1 . . . asn
Lecture 2
Math 407A: Linear Optimization


.
b1n



br 1 . . . brn
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
I
transportation scheduling
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
I
transportation scheduling
I
facility location
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
I
transportation scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
I
transportation scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
flight crew scheduling
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
resource allocation
I
production scheduling
I
warehousing
I
layout design
I
transportation scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
I
flight crew scheduling
I
portfolio optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
layout design
I
transportation scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
I
layout design
currency exchange
arbitrage
I
transportation scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
I
layout design
currency exchange
arbitrage
I
transportation scheduling
I
crop scheduling
I
facility location
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
I
layout design
currency exchange
arbitrage
I
transportation scheduling
I
crop scheduling
I
facility location
I
diet balancing
I
supply chain management
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
I
layout design
currency exchange
arbitrage
I
transportation scheduling
I
crop scheduling
I
facility location
I
diet balancing
I
supply chain management
I
parameter estimation
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington
Outline Gaussian Elimination Matrices Gauss-Jordan Elimination (Pivoting) What is linear programming? Applications of Line
Applications of Linear Programing
A short list:
I
flight crew scheduling
I
resource allocation
I
portfolio optimization
I
production scheduling
I
cash flow matching
I
warehousing
I
I
layout design
currency exchange
arbitrage
I
transportation scheduling
I
crop scheduling
I
facility location
I
diet balancing
I
supply chain management
I
parameter estimation
I
...
Lecture 2
Math 407A: Linear Optimization
Math Dept, University of Washington