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Chapter 5. Section 1. Theorem 5.1.1: If x and y are two nonzero vectors in either R 2 or R 3 and θ is the angle between them, then x T y =|| x |||| y ||cosθ. y -x. x. y. θ. Example. The projection of x onto y. x. y. θ. The projection of x onto y. x. y. θ. - PowerPoint PPT Presentation
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Chapter 5
Section 1
Theorem 5.1.1: If x and y are two nonzero vectors in either R2 or R3 and θ is the angle between them, then xTy=||x||||y||cosθ
x
y
y-x
θ
Example
x
yθ
The projection of x onto y
x
yθ
The projection of x onto y
x
yθ
The projection of x onto y
u
€
=1
|| r y ||r y
x
yθ
The projection of x onto y
uαu
x
yθ
The projection of x onto y
uαu
x-αu
x
yθ
The projection of x onto y
αu
x-αu
x
yθ
The projection of x onto y
αu
x-αu
x
yθ
The projection of x onto y
αu
x-αu
x
yθ
The projection of x onto y
αu
x-αu
Example
Example
Example
Example
y=(1,2)T
x=(5,2)T
Equations of Planes
Extensions to Rn:
€
|| x ||= (xT x)12 = (x1
2 + x22 + ...+ xn
2)12
|| x − y ||= ((x − y)T (x − y))12 = ((x1 − y1)
2 + ...+ (xn − yn )2)12
cosθ = xT y|| x ||⋅ || y ||
xT y = 0
Scalar Product:
Distance:
Angle:
Orthogonal when:
Extensions to Rn:
€
|| x ||= (xT x)12 = (x1
2 + x22 + ...+ xn
2)12
|| x − y ||= ((x − y)T (x − y))12 = ((x1 − y1)
2 + ...+ (xn − yn )2)12
cosθ = xT y|| x ||⋅ || y ||
xT y = 0
Scalar Product:
Distance:
Angle:
Orthogonal when:
Extensions to Rn:
€
|| x ||= (xT x)12 = (x1
2 + x22 + ...+ xn
2)12
|| x − y ||= ((x − y)T (x − y))12 = ((x1 − y1)
2 + ...+ (xn − yn )2)12
cosθ = xT y|| x ||⋅ || y ||
xT y = 0
Scalar Product:
Distance:
Angle:
Orthogonal when:
Extensions to Rn:
€
|| x ||= (xT x)12 = (x1
2 + x22 + ...+ xn
2)12
|| x − y ||= ((x − y)T (x − y))12 = ((x1 − y1)
2 + ...+ (xn − yn )2)12
cosθ = xT y|| x ||⋅ || y ||
xT y = 0
Scalar Product:
Distance:
Angle:
Orthogonal when:
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || x ||2
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || x ||2
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || x ||2
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || x ||2
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || y ||2
Vectors in Rn:
€
|| x + y ||2= (x + y)T (x + y)
€
=(xT + yT )(x + y)
= xT x + xT y + yT x + yT y
=|| x ||2 +2xT y+ || x ||2
If x and y are orthogonal…..
€
|| x + y ||2=|| x ||2 + || y ||2
x
y
x+y
B1: Applied Linear AlgebraB2: Elementary Linear AlgebraB3: Elementary Linear Algebra with
ApplicationsB4: Linear Algebra and its
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Keywords: algebra, application, elementary, linear, matrix, theory
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Keywords Book 1 Book 2 Book 3 Book 4 Book 5 Book 6 Book 7
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€
x =
110100
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A =
1 1 1 1 1 1 01 0 1 1 1 1 00 1 1 0 0 0 01 1 1 1 1 0 00 0 0 0 0 1 10 0 0 0 0 0 1
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6 × 7
Keywords Book 1 Book 2 Book 3 Book 4 Book 5 Book 6 Book 7
algebra 1 1 1 1 1 1 0
application 1 0 1 1 1 1 0
elementary 0 1 1 0 0 0 0
linear 1 1 1 1 1 0 0
matrix 0 0 0 0 0 1 1
theory 0 0 0 0 0 0 1
Search Engines
€
AT x =
1 1 0 1 0 01 0 1 1 0 01 1 1 1 0 01 1 0 1 0 01 1 0 1 0 01 1 0 0 1 00 0 0 0 1 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⋅
110100
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Keywords Book 1 Book 2 Book 3 Book 4 Book 5 Book 6 Book 7
algebra 1 1 1 1 1 1 0
application 1 0 1 1 1 1 0
elementary 0 1 1 0 0 0 0
linear 1 1 1 1 1 0 0
matrix 0 0 0 0 0 1 1
theory 0 0 0 0 0 0 1
Search Engines
€
AT x =
1 1 0 1 0 01 0 1 1 0 01 1 1 1 0 01 1 0 1 0 01 1 0 1 0 01 1 0 0 1 00 0 0 0 1 1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⋅
110100
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
3233320
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
B1: Applied Linear AlgebraB2: Elementary Linear AlgebraB3: Elementary Linear Algebra
with ApplicationsB4: Linear Algebra and its
applicationsB5: Linear Algebra with
ApplicationsB6: Matrix Algebra with
ApplicationsB7: Matrix Theory
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Key Words Doc 1 Doc 2 Doc 3 Doc 4 Doc 5 Doc 6 Doc 7 Doc 8
determinants 0 6 3 0 1 0 1 1
eigenvalues 0 0 0 0 0 5 3 2
linear 5 4 4 5 4 0 3 3
matrices 6 5 3 3 4 5 3 2
numerical 0 0 0 0 3 0 4 3
orthogonality 0 0 0 0 4 6 0 2
spaces 0 0 5 2 3 3 0 1
systems 5 3 3 2 4 2 1 1
transformations 0 0 0 5 1 3 1 0
vector 0 4 4 3 4 1 0 3
Relative Frequency Searches
Key Words Doc 1 Doc 2 Doc 3 Doc 4 Doc 5 Doc 6 Doc 7 Doc 8
determinants 0 6 3 0 1 0 1 1
eigenvalues 0 0 0 0 0 5 3 2
linear 5 4 4 5 4 0 3 3
matrices 6 5 3 3 4 5 3 2
numerical 0 0 0 0 3 0 4 3
orthogonality 0 0 0 0 4 6 0 2
spaces 0 0 5 2 3 3 0 1
systems 5 3 3 2 4 2 1 1
transformations 0 0 0 5 1 3 1 0
vector 0 4 4 3 4 1 0 3€
|| Doc1 ||
= 52 + 62 + 52
= 86
€
q1 = Doc1|| Doc1 ||
=
00
586
686
000
586
00
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
00
0.5390.647
000
0.53900
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Relative Frequency Searches
Key Words Doc 1 Doc 2 Doc 3 Doc 4 Doc 5 Doc 6 Doc 7 Doc 8
determinants 0 6 3 0 1 0 1 1
eigenvalues 0 0 0 0 0 5 3 2
linear 0.539 4 4 5 4 0 3 3
matrices 0.647 5 3 3 4 5 3 2
numerical 0 0 0 0 3 0 4 3
orthogonality 0 0 0 0 4 6 0 2
spaces 0 0 5 2 3 3 0 1
systems 0.539 3 3 2 4 2 1 1
transformations 0 0 0 5 1 3 1 0
vector 0 4 4 3 4 1 0 3
Relative Frequency Searches
€
Q =
0 .594 0.327 0 0.100 0 0.147 0.1540 0 0 0 0 .500 .442 0.309
.539 0.396 0.436 .574 0.400 0 .442 0.463
.647 0.495 0.327 0.344 0.400 0.400 .442 0.3090 0 0 0 0.300 0 .590 0.4630 0 0 0 0.400 0.600 0 0.3090 0 0.546 0.229 0.300 0.300 0 0.154
.539 .297 0.327 0.229 0.400 0.200 0.147 0.1540 0 0 0.574 0.100 0.300 0.147 00 0.396 0 0.344 0.400 0.100 0 0.463
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Relative Frequency Searches
Key Words Doc 1 Doc 2 Doc 3 Doc 4 Doc 5 Doc 6 Doc 7 Doc 8
determinants 0 6 3 0 1 0 1 1
eigenvalues 0 0 0 0 0 5 3 2
linear 0.539 4 4 5 4 0 3 3
matrices 0.647 5 3 3 4 5 3 2
numerical 0 0 0 0 3 0 4 3
orthogonality 0 0 0 0 4 6 0 2
spaces 0 0 5 2 3 3 0 1
systems 0.539 3 3 2 4 2 1 1
transformations 0 0 0 5 1 3 1 0
vector 0 4 4 3 4 1 0 3
Words to Search For:orthogonality
spacesvector
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x =
0000011001
⎡
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⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
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|| x ||= 3
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u =
00000
13
13
00
13
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
00000
0.5770.577
00
0.577
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Relative Frequency Searches Words to Search For:orthogonality
spacesvector
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QT u =
00.2290.5670.3310.6350.5770.577
00
0.577
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Note: Since y5=0.635 is the entry of y that is closest to 1, the direction of the search vector u is closest to the direction of q5 and hence document 5 is the one that best matches our search criteria
