Linear Algebra(Aljabar Linier)

Week 10

Universitas Multimedia NusantaraSerpong, Tangerang

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

Agenda

• Orthogonality– Orthogonality in Rn , Orthogonal complements, Orthogonal

Projections

– The Gram-Schmidt Process

– The QR Factorization• Approximating eigenvalues

– Orthogonal Diagonalization of Symmetric Matrices

• Vector Spaces– Vector spaces and subspaces

– Linear Independence, basis, and dimension

– Change of basis

– Linear Transformation: Kernel and Range

– The Matrix of a linear transformation

The Gram-Schmidt Processand

The QR Factorization

Constructing Orthogonal Vectors

•

The Gram-Schmidt Process

QR Factorization

• The Gram-Schmidt process has shown that for each i=1,...,n,

Example

• QR Factorization procedure:• Use the Gram-Schmidt process to find an orthonormal basis for Col A• Since Q has orthonormal columns, then . If then

• Find a QR factorization of

111

011

001

A

IQQ T QRA AQR T

Approximating Eigenvalues

• The idea is based on the following:

where

• All the Ak are similar and hence they have the same eigenvalues. Under certain conditions, the matrices Ak converge to a triangular matrix (the Schur form of A), where the eigenvalues are listed on the diagonal

• Example: Compute eigenvalues of

12

32A

kkkkkTkkkk

Tkkkk QAQQAQQRQQQRA 1

1

AA :0

Orthogonal Diagonalizationof Symmetric Matrices

Spectral Theorem

The spectral decomposition of AThe projection form of the Spectral Theorem

Example

• Find the spectral decomposition of the matrix

211

121

112

A

Vector Spaces&

Subspaces

• Definition:  Let V be a set on which addition and scalar multiplication are defined. If the following axioms are true for all objects u, v, and w in V and all scalars c and d then V is called a vector space and the objects in V are called “vectors”

• Note: objects called vectors here are not only Euclidean vectors (previous lectures), but they can be matrices, functions, etc.

Vector Spaces

• Let the set V be the points on a line through the origin, with the standard addition and scalar multiplication.  Show that V is a vector space .

• Let the set V be the points on a line that does NOT go through the origin in  with the standard addition and scalar multiplication.  Show that V is not a vector space

• Let n and m be fixed numbers and let   represent the set of nxm all  matrices.  Also let addition and scalar multiplication on   be the standard matrix addition and standard matrix scalar multiplication.   Show that   is a vector space

• Show that is a vector space:

Examples

nmMnmM

nmM

•

Operations on real-valued functions

• Theorem:

• Note: Every vector space, V, has at least two subspaces.  Namely, V itself and  (the zero space)

Subspaces

• Let W be the set of diagonal matrices of size nxn. Is this a subspace of Mnn ?

• Let   be the set of all polynomials of degree n or less. Is this a subspace of , where is a set of real-valued functions on the interval ?

Examples

nP],[ baF ],[ baF

],[ ba

Examples:

Spanning Sets

Linear IndependenceBasis

Dimension

Examples:

Linear Independence

Examples:

Basis

Remark: The most useful aspect pf coordinate vectors is that they allow us to transfer information from a general vector space to Rn

Examples:

Coordinates

Examples:

Dimension

Change of Basis

Introduction

The End

Thank you for your attention!