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

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

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

4. 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

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

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

7. Linear Algebra: Vectors
Spanning set:

8. Linear Algebra: Vectors
Basis:

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

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

11. Linear Algebra: Matrices
Basic matrix operations:
Addition
Multiplication

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

13. Linear Algebra: Matrices
Basic matrix operations:
Inversion

14. Linear Algebra: Matrices
Basic matrix operations:
Elementary row operations

15. 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?

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

17. 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

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

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

20. Convex Sets
Definition:
Prove convexity of:

21. Extreme Points
Definition:

22. Hyperplane and Half-space
Definition:

23. Rays and Directions
Definition:
Directions of a convex set:

24. Directions of A Convex Set
Polyhedral set directions:

25. Directions of A Convex Set
Polyhedral set directions:

26. Directions of A Convex Set
Polyhedral set directions:

27. Convex Functions
Definition:

28. Polyhedral Sets
Definition:
First inequality is redundant

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

30. Polyhedral Set Representation

31. Next time…
Simplex method