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inner product
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Some Applications
Inner Product, Length, Orthogonality
Orthogonalization
Least Squares Problems
Inner Product Spaces
Orthogonality and Least Squares
Least-Sqaures Problem
Some Applications
b Ax b Ax
Least-Sqaures Lines (Linear Regression)
Some Applications
Least-Sqaures polynomials (Polynomial Regression)
Some Applications
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
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
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
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
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
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
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
Some Applications of Fourier Series
3. Signal Processing and Filter Design
Some Applications
Stereo Equalizer
Some Applications of Fourier Series
4. Analysis of Electric Circuits
Some Applications
Some Applications of Fourier Series
5. Solution of PDE
Vibrating String
Some Applications
2 22
2 2
y yc
x t
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
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
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
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
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
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
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
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
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
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)
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
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
Theorem
For a matrix A,
• (Row A)=
• (Col A)=
Inner Product, Length, Orthogonality
Nul A
Nul AT
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
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
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
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