Linear Algebra(Aljabar Linier)

Week 5

Universitas Multimedia NusantaraSerpong, Tangerang

Dr. Ananda Kusumae-mail: ananda_kusuma@yahoo.com

Ph: 081338227031, 081908058069

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

Review

• Find a PTLU 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

412

321

600

A

132

213

211

a

a

a

10

01

00

034

432

212

A

0

1

3

b

SubspacesBasis

DimensionRank

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

Examples

• Show that the set of all vectors that satisfy the conditions x=3y

and z=-2y forms a subspace of R3

• Determine whether the set of all vectors , where y=x2, is a

subspace of R2

z

y

x

y

x

Subspaces associated with Matrices

Example:

Subspaces associated with Matrices (2)

Basis

• Find a basis for the row space, the column space and null space of

Dimension

Rank

Coordinates

Fundamental Theorem of Invertible Matrices

Version 2

Linear Transformations

Introduction

• A transformation (or mapping or function) T from Rn to Rm ( T: RnRm) is a rule that assigns to each vector v in Rn a unique vector T(v) in

Rm.• The domain of T is Rn, and the codomain of T is Rm.• 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 TA

– Image of and the range of TA

Linear Transformation

• Examples:– Let F:R2R2 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:RnRn such that I(v)=v for every v in Rn

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 v1 and v2?

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

The End

Thank you for your attention!