Linear Algebra (Aljabar Linier) Week 6 Universitas Multimedia Nusantara Serpong, Tangerang Dr. Ananda Kusuma Ph: ,

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

Linear Transformations

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

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.

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

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

Applications: Graphs

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.

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 ?

Digraph

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?

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

Eigenvalues & Eigenvectors

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:

Introduction Examples:

Determinants

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

Determinant 3x3 matrix

Determinant nxn matrix Example: Cofactor expansion along the first row

The Laplace Expansion Theorem Example:

Laplace Expansion: upper/lower triangular matrix Example:

Properties of Determinant Example:

Determinants of Elementary Matrices

Determinants and Matrix Operations

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:

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

The End Thank you for your attention!