linear algebra (aljabar linier) week 13 universitas multimedia nusantara serpong, tangerang dr....
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Linear Algebra(Aljabar Linier)
Week 13
Universitas Multimedia NusantaraSerpong, Tangerang
Dr. Ananda Kusumae-mail: [email protected]
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Agenda
• Review• Distance and Approximation
– Inner product spaces, norms and distance functions– Least Squares approximation
• Quiz
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1)
Review
2,1
2
1,1
2
1xxx
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Inner Product SpacesNorms and Distance Functions
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Length, Distance and Orthogonality
Inner Product Space
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a)
Examples
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Examples:
a) Sum norm:
b) Max norm:
c) P-norm:
d) 2-Norm or Euclidean norm:
Normed Vector Space
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Least Squares Approximation
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Motivation:
Curve of best fit
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Best Approximation
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- 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
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Least Square Theorem
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Find the parabola that gives the best least-squares approximation to the points (-1,1), (0,-1),(1,0), and (2,2)
Example
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The End
Thank you for your attention!