Linear Algebra (Aljabar Linier) Week 10 Universitas Multimedia Nusantara Serpong, Tangerang Dr. Ananda Kusuma

Agenda Orthogonality –Orthogonality in R n, 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 Process and The QR Factorization

Constructing Orthogonal Vectors

The Gram-Schmidt Process

Example Apply the Gram-Schmidt Process to construct an orthonormal basis for the subspace W = span(x 1,x 2,x 3 ) of R 4, where

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

Approximating Eigenvalues The idea is based on the following: where All the A k are similar and hence they have the same eigenvalues. Under certain conditions, the matrices A k converge to a triangular matrix (the Schur form of A), where the eigenvalues are listed on the diagonal Example: Compute eigenvalues of

Orthogonal Diagonalization of Symmetric Matrices

Spectral Theorem The spectral decomposition of A The projection form of the Spectral Theorem

Example Find the spectral decomposition of the matrix

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

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 M nn ? 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

Examples: Spanning Sets

Linear Independence Basis 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 R n Examples: Coordinates

Examples: Dimension

Change of Basis

Introduction

The End Thank you for your attention!