Download - Lecture16_Inner1
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Some Applications
Inner Product, Length, Orthogonality
Orthogonalization
Least Squares Problems
Inner Product Spaces
Orthogonality and Least Squares
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Least-Sqaures Problem
Some Applications
b Ax b Ax
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Least-Sqaures Lines (Linear Regression)
Some Applications
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Least-Sqaures polynomials (Polynomial Regression)
Some Applications
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
x
f x , x f x 2 f x2
French Scientist Jean-Baptiste Joseph Fourier
1768 - 1830
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
0n n
n 1
af x a cos nx b sin nx
2
x
f x , x f x 2 f x2
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
n 1
n 1
1f x sin nx
n
x
f x , x f x 2 f x2
sin x
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
n 1
n 1
1f x sin nx
n
x
f x , x f x 2 f x2
12
sin x sin 2x
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
n 1
n 1
1f x sin nx
n
x
f x , x f x 2 f x2
1 12 3
sin x sin 2x sin 3x
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Fourier Series
Any wave can be expressed as a series of sines and cosines
Some Applications
x
f x , x f x 2 f x2
n 1
n 1
1f x sin nx
n
http://upload.wikimedia.org/wikipedia/commons/e/e8/Periodic_identity_function.gif
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Some Applications of Fourier Series
1. Approximation Theory
2. Data compression
Some Applications
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Am
pli
tud
e
Frequency Index
Frequncy Spectrum
n 1
n 1
1f x sin nx
n
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Some Applications of Fourier Series
3. Signal Processing and Filter Design
Some Applications
Stereo Equalizer
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Some Applications of Fourier Series
4. Analysis of Electric Circuits
Some Applications
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Some Applications of Fourier Series
5. Solution of PDE
Vibrating String
Some Applications
2 22
2 2
y yc
x t
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Some Applications of Fourier Series
5. Solution of PDE
Vibrating String
Some Applications
2 22
2 2
y yc
x t
https://www.youtube.com/watch?v=9L9AOPxhZwY
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Inner Product in Rn
For U, V Rn,
The inner (dot) product is defined as:
1 1
2 2
n n
u v
u vU ,V
u v
Inner Product, Length, Orthogonality
T1 1 2 2 n nU V U V u v u v u v
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Ex. Find U·V, U·W,
1 2 2
2 3 3U ,V ,W
1 1 1
3 1 1
Inner Product, Length, Orthogonality
U V 1 2 2 3 1 1 3 1 2
U W 1 2 2 3 1 1 3 1 0
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Theorem
For U, V, W Rn and c R,
1. V·U =
2. (U+V)·W =
3. (cU)·V=
4. U·U
Inner Product, Length, Orthogonality
U V
U W V W
U (cV) c(U V)
0 ,U U 0 if and only if U 0
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Length (norm) of a Vector For U Rn ,
The length of U is defined as: 22 2 2 T
1 2 nU U U u u u and U U U U U
1
2
n
u
uU
u
Inner Product, Length, Orthogonality
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Unit Vector A unit vector is a vector whose length is 1
The unit vector in direction of V,
Ex. Find the unit vector in direction of V,
VU
V
1
2V
4
2
Inner Product, Length, Orthogonality
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Note that
For a scalar c,
15
25
45
25
1
2V 1U
4V 5
2
161 4 425 25 25 25U 1
Inner Product, Length, Orthogonality
V 1 4 16 4 5
cV c V
1
2U
4
2
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Distance Between Two Vectors
The distance between the U, V Rn, is the length of the
vector U-V, that is
dist U,V U V
Inner Product, Length, Orthogonality
2 2 2
1 1 2 2 n nu v u v u v
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Orthogonal Vectors U and V are orthogonal, if and only if
Note that, in R2 and R3,
dist U,V dist U, V
Inner Product, Length, Orthogonality
2 2
dist U,V dist U, V
U V U V U V U V
U V 0
U Vcos
U V
90 if U V 0
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Theorem For U and V Rn, the following statements are equivalent,
• U and V are orthogonal
• U·V = 0
• dist(U,V) = dist(U,-V)
• ǁU+Vǁ2 = ǁUǁ2 + ǁVǁ2
Inner Product, Length, Orthogonality
(Pythagorean Theorem)
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Ex. Are U and V orthogonal?
U and V are orthogonal
Inner Product, Length, Orthogonality
U V 4 5 3 6 2 1 0
4 5
U 3 ,V 6
2 1
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Orthogonal Complement • If a vector z is orthogonal to every vector in a subspace
W of Rn, then z is said to be orthogonal to W
• The set of all vectors z that are orthogonal to W is called the orthogonal complement of W and is denoted by W
Ex. For a plane W passing through the
origin in R3, and the line L passing
through the origin and orthogonal to W
W = L
L = W
Inner Product, Length, Orthogonality
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Theorem
For a matrix A,
• (Row A)=
• (Col A)=
Inner Product, Length, Orthogonality
Nul A
Nul AT
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Orthogonal Set A set of nonzero vectors S={V1,V2,…Vp} in Rn is an orthogonal set if Vi·Vj =0, ij
Ex. IS S={V1,V2,V3} an orthogonal set?
V1·V2 = V1·V3 = V2·V3 = 0
S is an orthogonal set
Inner Product, Length, Orthogonality
1 2 3
3 1 3
2 3 8V ,V ,V
1 3 7
3 4 0
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Orthonormal Set A set of nonzero vectors S={U1,U2,…Up} in Rn is an orthonormal set if Ui·Uj =0, ij and ǁUiǁ=1 for all i
Ex. IS S={U1,U2,U3} an orthonormal set?
U1·U2 = U1·U3 = U2·U3 = 0, ǁU1ǁ=ǁU2ǁ=ǁU3ǁ=1
S is an orthonormal set
Inner Product, Length, Orthogonality
1 2 23 3 3
2 2 11 2 33 3 3
2 1 23 3 3
U ,U ,U
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Orthogonal Basis If S={V1,V2,…Vp} is an orthogonal set of nonzero vectors in Rn, then S is linearly independent (orthogonal basis for Span{S})
Orthonormal Basis If S={U1,U2,…Up} is an orthonormal set of nonzero vectors in Rn, then S is linearly independent (orthonormal basis for Span{S})
Inner Product, Length, Orthogonality
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Ex. Given the following vectors,
a. Are V1,V2,V3 linearly independent?
b. Do V1,V2,V3 form a basis for R3?
1 2 3
2 0 0
V 0 ,V 3 ,V 0
0 0 4
Yes
Yes (orthogonal basis)
Inner Product, Length, Orthogonality