EMGT 6412/MATH 6665
Mathematical Programming
Spring 2016
Linear Algebra/Sets Review
Dincer Konur
Engineering Management and Systems
Engineering
1
Outline
• Linear Algebra
– Vectors
– Matrices
– Linear System
• Sets
–
–
–
–
–
Convex Sets
Extreme points and hyperplanes
Directions
Polyhedral sets
Representation
Chapter 2
2
Outline
• Linear Algebra
– Vectors
– Matrices
– Linear System
• Sets
–
–
–
–
–
Convex Sets
Extreme points and hyperplanes
Directions
Polyhedral sets
Representation
3
Linear Algebra: Vectors
• An n-vector is a row or column array of n numbers
• Zero vector, 0, all zeros
• ith unit vector, ei, ith component
is 1, others are 0
• Sum vector, 1, has all ones
– Addition:
– Inner product:
•
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Linear Algebra: Vectors
• Linear and affine combinations:
• Linear Independence:
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Linear Algebra: Vectors
• Linear Independence:
–
–
– Then
–
and
are linearly independent
–
–
– Then these vectors are linearly dependent
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Linear Algebra: Vectors
• Spanning set:
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Linear Algebra: Vectors
• Basis:
8
Linear Algebra: Vectors
• Replacing a vector from Basis with another one:
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Linear Algebra: Vectors
• Replacing a vector from Basis with another one:
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Linear Algebra: Matrices
• Basic matrix operations:
– Addition
– Multiplication
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Linear Algebra: Matrices
• Basic matrix operations:
– Transposition
– Special matrices
• Zero matrix
• Identity matrix
• Triangular matrix
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Linear Algebra: Matrices
• Basic matrix operations:
– Inversion
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Linear Algebra: Matrices
• Basic matrix operations:
– Elementary row operations
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Linear Algebra: Matrices
• Basic matrix operations:
– Rank of a matrix
• It can be shown that the row rank of a matrix is always equal to its
column rank, and hence the rank of the matrix is equal to the
maximum number of linearly independent rows (or columns) of A.
• Thus it is clear that rank (A) <=minimum {m, n}.
• If rank (A) = minimum {m, n}, A is said to be of full rank.
– Practice: how to find the rank of a matrix?
• http://stattrek.com/matrix-algebra/matrix-rank.aspx
• http://stattrek.com/matrix-algebra/echelon-transform.aspx#MatrixA
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Linear Algebra: Linear System
• Consider a system of linear equations:
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Linear Algebra: Linear System
• Consider a system of linear equations:
–
–
–
–
•
•
•
B exists since
• B is called a basis matrix (since the columns of B form a basis of R )
• N is called the corresponding nonbasic matrix
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Linear Algebra: Linear System
• Consider a system of linear equations:
–
•
•
•
• Since B has inverse,
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Outline
• Linear Algebra
– Vectors
– Matrices
– Linear System
• Sets
–
–
–
–
–
Convex Sets
Extreme points and hyperplanes
Directions
Polyhedral sets
Representation
19
Convex Sets
• Definition:
–
– Prove convexity of:
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Extreme Points
• Definition:
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Hyperplane and Half-space
• Definition:
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Rays and Directions
• Definition:
• Directions of a convex set:
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Directions of A Convex Set
• Polyhedral set directions:
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Directions of A Convex Set
• Polyhedral set directions:
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Directions of A Convex Set
• Polyhedral set directions:
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Convex Functions
• Definition:
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Polyhedral Sets
• Definition:
First inequality is
redundant
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Polyhedral Sets
• Representation:
– Including x>=0, there are (m+n) defining half-spaces
–
–
–
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Polyhedral Set Representation
30
Next time…
• Simplex method
31