1. Linear Algebra (Aljabar Linier) Week 5 Universitas Multimedia Nusantara Serpong, Tangerang Dr. Ananda Kusuma e-mail: ananda_kusuma@yahoo.com Ph: 081338227031, 081908058069

2. Agenda Exercises on matrix operation, matrix inversion, and LU factorization Matrices –Matrix Operations and Algebra –Matrix Inversion –LU Factorization –Subspaces, Basis, Dimension and Rank –Linear Transformations –Applications Graph Markov Chains

3. Review Find a P T LU factorization of, where P is a permutation matrix  Adapting the LU factorization to handle cases where row interchanges are necessary during Gaussian elimination Solve X for : AXB = (BA)2,ABXA-1B-1 = I + A Using Gauss-Jordan method, find the inverse of the given matrix (if it exists) Solve the system Ax=b using LU factorization, where

4. Subspaces Basis Dimension Rank

5. Subspaces A subspace is a subset of vector space (Lecture Week 10). –To learn subspaces relating to matrices in order to understand the structure of Ax=b

6. Examples Show that the set of all vectors that satisfy the conditions x=3y and z=-2y forms a subspace of R 3 Determine whether the set of all vectors, where y=x 2, is a subspace of R 2

7. Subspaces associated with Matrices Example:

8. Subspaces associated with Matrices (2)

9. Basis Find a basis for the row space, the column space and null space of

10. Dimension

11. Rank

12. Coordinates

13. Fundamental Theorem of Invertible Matrices Version 2

14. Linear Transformations

15. Introduction A transformation (or mapping or function) T from R n to R m ( T: R n  R m ) is a rule that assigns to each vector v in R n a unique vector T(v) in R m. The domain of T is R n, and the codomain of T is R m. For a vector v in the domain of T, the vector T(v) in the codomain is called the image of v under the action of T. The set of all possible images T(v) (as v varies throughout the domain of T) is called the range of T Example:  Find –Domain and codomain of T A –Image of and the range of T A

16. Linear Transformation Examples: –Let F:R 2  R 2 be the transformation that sends each point to its reflection in the x-axis. Show that F is a linear transformation.

17. Composition of Linear Transformations Example:  Consider the following linear transformation T and S. Find

18. Inverse of Linear Transformations Where I is an identity transformation, I:R n  R n such that I(v)=v for every v in R n

19. Applications: Graphs

20. Adjacency Matrix In week 3 we studied network analysis which in essence is the application of graph. We can record the essential information about a grah in a matrix, and use matrix algebra to answer certain questions about the graph.

21. Path A path in a graph is a sequence of edges from one vertex other vertex. The length of a path is the number of edges it contains, and we will refer to a path with k edges as a k-path How many 3-paths are there between v 1 and v 2 ?

22. Digraph

23. Tournament Five tennis players (Davenport, Graf, Hingis, Seles and Williams) compete in a round-robin tournament in which each player plays every other player once. A directed edge from vertex i to vertex j means player i defeated player j Tournament  a directed graph (digraph) in which there is exactly one directed edge between every pair of vertices. How to rank the players?

24. Ranking Count the number of wins for each player: Count indirect wins  2-path in the digraph Ranking: Davenport, Graf, Hingis, Williams, Seles

25. The End Thank you for your attention!