Linear Algebra(Aljabar Linier)

Week 13

Universitas Multimedia NusantaraSerpong, Tangerang

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

Agenda

• Review• Distance and Approximation

– Inner product spaces, norms and distance functions– Least Squares approximation

• Quiz

1)

Review

2,1

2

1,1

2

1xxx

Inner Product SpacesNorms and Distance Functions

Length, Distance and Orthogonality

Inner Product Space

a)

Examples

Examples:

a) Sum norm:

b) Max norm:

c) P-norm:

d) 2-Norm or Euclidean norm:

Normed Vector Space

Least Squares Approximation

Motivation:

Curve of best fit

Best Approximation

- Minimizing total error

- Example: n data points and a line

- The error vector is

where

- We can construct a system of linear equation in matrix form , where

- The error vector

Least Square for Curve Fitting

222

21

2

n e

nn yxyxyx ,,,,,, 2211 bxay

n

1

e)( iii bxay

bA x

xAbe

Least Square Theorem

Find the parabola that gives the best least-squares approximation to the points (-1,1), (0,-1),(1,0), and (2,2)

Example

The End

Thank you for your attention!