1. Linear Algebra (Aljabar Linier) Week 6 Universitas Multimedia Nusantara Serpong, Tangerang Dr. Ananda Kusuma e-mail: ananda_kusuma@yahoo.com Ph: 081338227031, 081908058069

2. Agenda Continue on Last Week’s lecture –Linear Transformations –Applications Graph Markov Chains Eigenvalues and Eigenvectors –Introduction –Determinants –Eigenvalues of nxn Matrices –Similarity and Diagonalization –Iterative methods for computing eigenvalues –Applications Graphs: search engines Markov Chains

3. Linear Transformations

4. 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

5. 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.

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

7. 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

8. Applications: Graphs

9. 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.

10. 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 ?

11. Digraph

12. 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?

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

14. Eigenvalues & Eigenvectors

15. Introduction Iterative process or dynamical systems may reach to steady states in certain situation. E.g.: Markov Chain, Leslie model of population growth, etc. Markov Chain: –Evolving process consisting of a finite number of states. It satisfies –Where x k is state vector and P is called transition matrix (containing transition probabilities) –Completely determined by its transition probabilities and its initial state –Example:

16. Introduction Examples:

17. Determinants

18. Introduction Recall the formula for computing the inverse a matrix: Let matrix, The inverse: The determinant: Let matrix The inverse:

19. Determinant 3x3 matrix

20. Determinant nxn matrix Example: Cofactor expansion along the first row

21. The Laplace Expansion Theorem Example:

22. Laplace Expansion: upper/lower triangular matrix Example:

23. Properties of Determinant Example:

24. Determinants of Elementary Matrices

25. Determinants and Matrix Operations

26. Cramer’s Rule and the Adjoint Use of determinants to finding solution of linear systems and the inverse of matrices They are not practical (computationally inefficient), but they are of great theoretical importance It is more efficient to use procedure such as Gaussian elimination to solve the system directly where Example:

27. Using Adjoint to compute inverse of a matrix where Example: Transpose of the matrix of cofactors

28. The End Thank you for your attention!