EMGT 6412/MATH 6665 Mathematical Programming Spring 2016 Linear Algebra/Sets Review Dincer Konur Engineering Management and Systems Engineering

Outline Chapter 2 Linear Algebra Sets Vectors Matrices Linear System Convex Sets Extreme points and hyperplanes Directions Polyhedral sets Representation Chapter 2

Outline Linear Algebra Sets Vectors Matrices Linear System Convex Sets Extreme points and hyperplanes Directions Polyhedral sets Representation

Linear Algebra: Vectors An n-vector is a row or column array of n numbers Addition: Inner product: Zero vector, 0, all zeros ith unit vector, ei, ith component is 1, others are 0 Sum vector, 1, has all ones

Linear Algebra: Vectors Linear and affine combinations: Linear Independence:

Linear Algebra: Vectors Linear Independence: Then and are linearly independent Then these vectors are linearly dependent

Linear Algebra: Vectors Spanning set:

Linear Algebra: Vectors Basis:

Linear Algebra: Vectors Replacing a vector from Basis with another one:

Linear Algebra: Vectors Replacing a vector from Basis with another one:

Linear Algebra: Matrices Basic matrix operations: Addition Multiplication

Linear Algebra: Matrices Basic matrix operations: Transposition Special matrices Zero matrix Identity matrix Triangular matrix

Linear Algebra: Matrices Basic matrix operations: Inversion

Linear Algebra: Matrices Basic matrix operations: Elementary row operations

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

Linear Algebra: Linear System Consider a system of linear equations:

Linear Algebra: Linear System Consider a system of linear equations: B is called a basis matrix (since the columns of B form a basis of R ) N is called the corresponding nonbasic matrix B exists since

Linear Algebra: Linear System Consider a system of linear equations: Since B has inverse,

Outline Linear Algebra Sets Vectors Matrices Linear System Convex Sets Extreme points and hyperplanes Directions Polyhedral sets Representation

Convex Sets Definition: Prove convexity of:

Extreme Points Definition:

Hyperplane and Half-space Definition:

Rays and Directions Definition: Directions of a convex set:

Directions of A Convex Set Polyhedral set directions:

Directions of A Convex Set Polyhedral set directions:

Directions of A Convex Set Polyhedral set directions:

Convex Functions Definition:

Polyhedral Sets Definition: First inequality is redundant

Polyhedral Sets Representation: Including x>=0, there are (m+n) defining half-spaces

Polyhedral Set Representation

Next time… Simplex method