The document discusses vectors and inner product spaces. It defines key concepts such as the length of a vector, the dot product, orthogonality of vectors, and properties of inner products. It also introduces the concepts of an inner product space and orthogonal projections. The orthogonal projection of a vector onto another vector is defined as the projection of the first vector onto the direction of the second vector.
Second Part of Inner Products including Cauchy Schwarz inequality, Pythagorean theorem, Orthogonal complement, Projection theorem with geometrical interpretation, Least Square Solution
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Second Part of Inner Products including Cauchy Schwarz inequality, Pythagorean theorem, Orthogonal complement, Projection theorem with geometrical interpretation, Least Square Solution
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Hankel Determinent for Certain Classes of Analytic FunctionsIJERA Editor
Let 1 A denote the class of functions
2
( )
n
n
n f z z a z analytic in the unit disc E {z : |z| <1}.
M denotes the class of functions in 1 A which satisfy the conditions 0
( ). ( )
z
f z f z
and for
0 1, 0
( )
( ( ))
( )
( )
) 1 ( Re
f z
zf z
f z
zf z
. We are interested in determining the sharp upper bound
for the functional
2
2 4 3 a a a for the class M .
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
On α-characteristic Equations and α-minimal Polynomial of Rectangular MatricesIOSR Journals
In this paper, we study rectangular matrices which satisfy the criteria of the Cayley-Hamilton
theorem for a square matrix.Various results on characteristic polynomials, characteristic equations,
eigenvalues and α-minimal polynomial of rectangular matrices are proved.
AMS SUBJECT CLASSIFICATION CODE: 17D20(γ,δ).
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Hankel Determinent for Certain Classes of Analytic FunctionsIJERA Editor
Let 1 A denote the class of functions
2
( )
n
n
n f z z a z analytic in the unit disc E {z : |z| <1}.
M denotes the class of functions in 1 A which satisfy the conditions 0
( ). ( )
z
f z f z
and for
0 1, 0
( )
( ( ))
( )
( )
) 1 ( Re
f z
zf z
f z
zf z
. We are interested in determining the sharp upper bound
for the functional
2
2 4 3 a a a for the class M .
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
On α-characteristic Equations and α-minimal Polynomial of Rectangular MatricesIOSR Journals
In this paper, we study rectangular matrices which satisfy the criteria of the Cayley-Hamilton
theorem for a square matrix.Various results on characteristic polynomials, characteristic equations,
eigenvalues and α-minimal polynomial of rectangular matrices are proved.
AMS SUBJECT CLASSIFICATION CODE: 17D20(γ,δ).
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
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How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
3. 5.1 Length and Dot Product in Rn
5.2 Inner Product Spaces
5.3 Orthonormal Bases:Gram-Schmidt Process
5.4 Mathematical Models and Least Square
Analysis
4. Length:
The length of a vector in Rn is
given by
22
2
2
1|||| nvvv v
Notes: Properties of length
vv
vv
vv
v
cc4
0iff03
12
01
is called a unit vector.
),,,( 21 nvvv v
Notes: The length of a vector is also called its norm.
5. • Ex 1:
(a) In R5, the length of is
given by
(b) In R3 the length of is given
by
)2,4,1,2,0(v
525)2(41)2(0|||| 22222
v
1
17
17
17
3
17
2
17
2
||||
222
v
),,( 17
3
17
2
17
2
v
(v is a unit vector)
6. A standard unit vector in Rn:
02
01
c
c
cvu
u and v have the same direction
u and v have the opposite direction
Notes: (Two nonzero vectors are parallel)
1,,0,0,0,,1,0,0,,0,1,,, 21 neee
Ex:
the standard unit vector in R2:
the standard unit vector in R3:
1,0,0,1, ji
1,0,0,0,1,0,0,0,1,, kji
7. Thm 5.1: (Length of a scalar multiple)
Let v be a vector in Rn and c be a scalar. Then
|||||||||| vv cc
||||||
||
)(
)()()(
||),,,(||||||
22
2
2
1
22
2
2
1
2
22
2
2
1
21
v
v
c
vvvc
vvvc
cvcvcv
cvcvcvc
n
n
n
n
Pf:
),,,( 21 nvvv v
),,,( 21 ncvcvcvc v
8. • Thm 5.2: (Unit vector in the direction of v)
If v is a nonzero vector in Rn, then the vector
has length 1 and has the same direction as v. This
vector u is called the unit vector in the direction of v.
|||| v
v
u
Pf:
v is nonzero 0
1
0
v
v
v
v
1
u (u has the same direction as v)
1||||
||||
1
||||
|||| v
vv
v
u (u has length 1 )
9. • Notes:
(1) The vector is called the unit vector in the
direction of v.
(2) The process of finding the unit vector in the
direction of v
is called normalizing the vector v.
|||| v
v
10. • Ex 2: (Finding a unit vector)
Find the unit vector in the direction of ,
and verify that this vector has length 1.
14
2
,
14
1
,
14
3
)2,1,3(
14
1
2)1(3
)2,1,3(
|||| 222v
v
1
14
14
14
2
14
1
14
3
222
v
v
is a unit vector.
)2,1,3(v 14213 222
v
Sol:
)2,1,3(v
11. Distance between two vectors:
The distance between two vectors u and v in Rn is
||||),( vuvud
Notes: (Properties of distance)
(1)
(2) if and only if
(3)
0),( vud
0),( vud vu
),(),( uvvu dd
12. • Ex 3: (Finding the distance between two vectors)
The distance between u=(0, 2, 2) and v=(2, 0, 1) is`
312)2(
||)12,02,20(||||||),(
222
vuvud
13. Dot product in Rn:
The dot product of and
is the scalar quantity
Ex 4: (Finding the dot product of two vectors)
The dot product of u=(1, 2, 0, -3) and v=(3, -2, 4, 2) is
7)2)(3()4)(0()2)(2()3)(1(vu
nnvuvuvu 2211vu
),,,( 21 nuuu u
),,,( 21 nvvv v
14. Thm 5.3: (Properties of the dot product)
If u, v, and w are vectors in Rn and c is a scalar,
then the following properties are true.
(1)
(2)
(3)
(4)
(5) , and if and only if
uvvu
wuvuwvu )(
)()()( vuvuvu ccc
2
|||| vvv
0vv 0vv 0v
15. • Euclidean n-space:
Rn was defined to be the set of all order n-tuples of
real numbers. When Rn is combined with the standard
operations of vector addition, scalar multiplication,
vector length, and the dot product, the resulting
vector space is called Euclidean n-space.
16. • Ex 5: (Using the properties of the dot product)
Given
39uu 3vu 79vv
)3()2( vuvu
Sol:
)3(2)3()3()2( vuvvuuvuvu
Find
254)79(2)3(7)39(3
vvuvvuuu )2()3()2()3(
)(2)(6)(3 vvuvvuuu
)(2)(7)(3 vvvuuu
17. • Ex 7: (An example of the Cauchy - Schwarz inequality)
Verify the Cauchy - Schwarz inequality for u=(1, -
1, 3)
and v=(2, 0, -1)
Thm 5.4: (The Cauchy - Schwarz inequality)
If u and v are vectors in Rn, then
( denotes the absolute value of )
|||||||||| vuvu || vu vu
vuvu
vvuuvu
vu
55511
11
5,11,1 vvuuvuSol:
18. The angle between two vectors in Rn:
Note:
The angle between the zero vector and another vector is not
defined.
0,
||||||||
cos
vu
vu
1cos 1cos
0
0cos
2
0cos
2
0cos
2
0
0vu 0vu 0vu
Opposit
e
directio
n
Same
directio
n
19. Ex 8: (Finding the angle between two vectors)
)2,2,0,4(u )1,1,0,2(v
Sol:
242204
2222
uuu
1
144
12
624
12
||||||||
cos
vu
vu
61102 2222
vvv
12)1)(2()1)(2()0)(0()2)(4(vu
u and v have opposite directions. )2( vu
20. Orthogonal vectors:
Two vectors u and v in Rn are orthogonal if
0vu
Note:
The vector 0 is said to be orthogonal to every vector.
21. Ex 10: (Finding orthogonal vectors)
Determine all vectors in Rn that are orthogonal to u=(4, 2).
0
24
),()2,4(
21
21
vv
vvvu
0
2
1
1024
tv
t
v 21 ,
2
Rt,t
t
,
2
v
)2,4(u Let ),( 21 vvv
Sol:
22. Thm 5.5: (The triangle inequality)
If u and v are vectors in Rn, then
|||||||||||| vuvu
Pf:
)()(|||| 2
vuvuvu
2222
||||||2||||||||)(2||||
)(2)()(
vvuuvvuu
vvvuuuvuvvuu
2
22
||)||||(||
||||||||||||2||||
vu
vvuu
|||||||||||| vuvu
Note:
Equality occurs in the triangle inequality if and only if
the vectors u and v have the same direction.
23. Thm 5.6: (The Pythagorean theorem)
If u and v are vectors in Rn, then u and v are orthogonal
if and only if
222
|||||||||||| vuvu
24. Dot product and matrix multiplication:
nu
u
u
2
1
u
nv
v
v
2
1
v
][][ 2211
2
1
21 nn
n
n
T
vuvuvu
v
v
v
uuu
vuvu
(A vector in Rn
is represented as an n×1 column matrix)
),,,( 21 nuuu u
25. • Inner product:
Let u, v, and w be vectors in a vector space V, and
let c be any scalar. An inner product on V is a
function that associates a real number <u, v> with
each pair of vectors u and v and satisfies the
following axioms.
(1)
(2)
(3)
(4) `if and only if
〉〈〉〈 uvvu ,,
〉〈〉〈〉〈 wuvuwvu ,,,
〉〈〉〈 vuvu ,, cc
0, 〉〈 vv 0, 〉〈 vv 0v
27. Ex 1: (The Euclidean inner product for Rn)
Show that the dot product in Rn satisfies the four axioms of an
inner product.
nnvuvuvu 2211, vuvu 〉〈
),,,(,),,,( 2121 nn vvvuuu vu
Sol:
By Theorem 5.3, this dot product satisfies the required four axioms. Thus
it is an inner product on Rn.
28. Ex 2: (A different inner product for Rn)
Show that the function defines an inner product on R2, where
and .
2211 2, vuvu〉〈 vu
),(),( 2121 vvuu vu
Sol:
〉〈〉〈 uvvu ,22,)( 22112211 uvuvvuvua
〉〈〉〈
〉〈
wuvu
wvu
,,
)2()2(
22
)(2)(,
22112211
22221111
222111
wuwuvuvu
wuvuwuvu
wvuwvu
),()( 21 wwb w
30. Ex 3: (A function that is not an inner product)
Show that the following function is not an inner product on R3.
332211 2 vuvuvu〉〈 vu
Sol:
Let
)1,2,1(v
06)1)(1()2)(2(2)1)(1(,Then vv
Axiom 4 is not satisfied.
Thus this function is not an inner product on R3.
31. Thm 5.7: (Properties of inner products)
Let u, v, and w be vectors in an inner product space V, and let c
be any real number.
(1)
(2)
(3)
0,, 〉〈〉〈 0vv0
〉〈〉〈〉〈 wvwuwvu ,,,
〉〈〉〈 vuvu ,, cc
Norm (length) of u:
〉〈 uuu ,||||
〉〈 uuu ,|||| 2
Note:
32. u and v are orthogonal if .
Distance between u and v:
vuvuvuvu ,||||),(d
Angle between two nonzero vectors u and v:
0,
||||||||
,
cos
vu
vu 〉〈
Orthogonal:
0, 〉〈 vu
)( vu
33. Notes:
(1) If , then v is called a unit vector.
(2)
1||||v
0
1
v
v gNormalizin
v
v (the unit vector in the
direction of v)
not a unit vector
35. Properties of norm:
(1)
(2) if and only if
(3)
Properties of distance:
(1)
(2) if and only if
(3)
0||||u
0||||u 0u
|||||||||| uu cc
0),( vud
0),( vud vu
),(),( uvvu dd
36. Thm 5.8:
Let u and v be vectors in an inner product space V.
(1) Cauchy-Schwarz inequality:
(2) Triangle inequality:
(3) Pythagorean theorem :
u and v are orthogonal if and only if
|||||||||||| vuvu
Theorem 5.5
222
|||||||||||| vuvu Theorem 5.6
|||||||||,| vuvu 〉〈 Theorem 5.4
37. Orthogonal projections in inner product spaces:
Let u and v be two vectors in an inner product space V, such
that . Then the orthogonal projection of u onto v is given
by 0v
v
vv
vu
uv
,
,
proj
Note:
If v is a init vector, then .
The formula for the orthogonal projection of u onto v takes the
following simpler form.
1||||, 2
vvv 〉〈
vvuuv ,proj
38. Ex 10: (Finding an orthogonal projection in R3)
Use the Euclidean inner product in R3 to find the orthogonal
projection of u=(6, 2, 4) onto v=(1, 2, 0).
Sol:
10)0)(4()2)(2()1)(6(, vu
5021, 222
vv
)0,4,2()0,2,1(proj 5
10
v
vv
vu
uv
Note:
).0,2,1(toorthogonalis4)2,(4,0)4,(2,4)2,(6,proj vuu v
39. Thm 5.9: (Orthogonal projection and distance)
Let u and v be two vectors in an inner product space V, such
that . Then
0v
vv
vu
vuuu v
,
,
,),()proj,( ccdd
40. • Orthogonal:
A set S of vectors in an inner product space V is called an
orthogonal set if every pair of vectors in the set is orthogonal.
Orthonormal:
An orthogonal set in which each vector is a unit vector is called
orthonormal.
ji
ji
VS
ji
n
0
1
,
,,, 21
vv
vvv
0,
,,, 21
ji
n VS
vv
vvv
ji
Note:
If S is a basis, then it is called an orthogonal basis or an
orthonormal basis.
41. • Ex 1: (A nonstandard orthonormal basis for R3)
Show that the following set is an orthonormal basis.
3
1
,
3
2
,
3
2
,
3
22
,
6
2
,
6
2
,0,
2
1
,
2
1
321
S
vvv
Sol:
Show that the three vectors are mutually orthogonal.
0
9
22
9
2
9
2
00
23
2
23
2
00
32
31
6
1
6
1
21
vv
vv
vv
42. Show that each vector is of length 1.
Thus S is an orthonormal set.
1||||
1||||
10||||
9
1
9
4
9
4
333
9
8
36
2
36
2
222
2
1
2
1
111
vvv
vvv
vvv
43. The standard basis is orthonormal.
Ex 2: (An orthonormal basis for )
In , with the inner product
)(3 xP
221100, bababaqp
},,1{ 2
xxB
)(3 xP
Sol:
,001 2
1 xxv ,00 2
2 xxv ,00 2
3 xxv
0)1)(0()0)(1()0)(0(,
,0)1)(0()0)(0()0)(1(,
,0)0)(0()1)(0()0)(1(,
32
31
21
vv
vv
vv
Then
45. • Thm 5.10: (Orthogonal sets are linearly
independent)
If is an orthogonal set of nonzero
vectors in an inner product space V, then S is linearly
independent.
nS v,,v,v 21
Pf:
S is an orthogonal set of nonzero vectors
0and0i.e. iiji ji v,vv,v
iccc
ccc
iinn
nn
0,0,
0Let
2211
2211
vvvvv
vvv
iii
inniiiii
c
cccc
v,v
v,vv,vv,vv,v 2211
t.independenlinearlyis00 Siciii v,v
46. Corollary to Thm 5.10:
If V is an inner product space of dimension n, then any
orthogonal set of n nonzero vectors is a basis for V.
47. Ex 4: (Using orthogonality to test for a basis)
Show that the following set is a basis for .4
R
)}1,1,2,1(,)1,2,0,1(,)1,0,0,1(,)2,2,3,2{(
4321
S
vvvv
Sol:
: nonzero vectors
02262
02402
02002
41
31
21
vv
vv
vv
4321 ,,, vvvv
01201
01001
01001
43
42
32
vv
vv
vv
.orthogonalisS
4
forbasisais RS (by Corollary to Theorem 5.10)
48. • Thm 5.11: (Coordinates relative to an orthonormal
basis)
If is an orthonormal basis for an
inner product space V, then the coordinate representation
of a vector w with respect to B is
},,,{ 21 nB vvv
},,,{ 21 nB vvv is orthonormal
ji
ji
ji
0
1
, vv
Vw
nnkkk vvvw 2211
(unique representation)
Pf:
is a basis for V
},,,{ 21 nB vvv
nn vvwvvwvvww ,,, 2211
50. • Ex 5: (Representing vectors relative to an
orthonormal basis)
Find the coordinates of w = (5, -5, 2) relative to the
following orthonormal basis for .
)}1,0,0(,)0,,(,)0,,{( 5
3
5
4
5
4
5
3
B
3
R
Sol:
2)1,0,0()2,5,5(,
7)0,,()2,5,5(,
1)0,,()2,5,5(,
33
5
3
5
4
22
5
4
5
3
11
vwvw
vwvw
vwvw
2
7
1
][ Bw
51. • Gram-Schmidt orthonormalization process:
is a basis for an inner product space V},,,{ 21 nB uuu
11Let uv })({1 1vw span
}),({2 21 vvw span
},,,{' 21 nB vvv
},,,{''
2
2
n
n
B
v
v
v
v
v
v
1
1
is an orthogonal basis.
is an orthonormal basis.
1
1 〉〈
〉〈
proj 1
n
i
i
ii
in
nnnn n
v
v,v
v,v
uuuv W
2
22
23
1
11
13
3333
〉〈
〉〈
〉〈
〉〈
proj 2
v
v,v
v,u
v
v,v
v,u
uuuv W
1
11
12
2222
〉〈
〉〈
proj 1
v
v,v
v,u
uuuv W
54. • Ex 10: (Alternative form of Gram-Schmidt orthonormalization process)
Find an orthonormal basis for the solution space of the
homogeneous system of linear equations.
0622
07
4321
421
xxxx
xxx
Sol:
08210
01201
06212
07011 .. EJG
1
0
8
1
0
1
2
2
82
2
4
3
2
1
ts
t
s
ts
ts
x
x
x
x
55. Thus one basis for the solution space is
)}1,0,8,1(,)0,1,2,2{(},{ 21
uuB
1,2,4,3
01,2,,2
9
18
10,8,1,
,
,
01,2,,2
1
11
12
22
11
v
vv
vu
uv
uv
1,2,4,30,1,2,2'B
(orthogonal basis)
30
1
,
30
2
,
30
4
,
30
3
,0,
3
1
,
3
2
,
3
2
''B
(orthonormal basis)
57. Let W be a subspace of an inner product
space V.
(a) A vector u in V is said to orthogonal to W,
if u is orthogonal to every vector in W.
(b) The set of all vectors in V that are orthogonal to
every
vector in W is called the orthogonal
complement of W.
(read “ perp”)
},0,|{ WVW wwvv
W W
Orthogonal complement of W:
0(2)0(1) VV
Notes:
59. Thm 5.14: (Projection onto a subspace)
If is an orthonormal basis for
the subspace S of V, and , then
),0,proj(,, 11
iiWtt
uvuuvuuv
},,,{ 21 t
uuu
Vv
ttW uuvuuvuuvv ,,,proj 2211
Pf:
for Wbasislorthonormaanis},,,{andWproj 21 tW
uuuv
ttWWW
uuvuuvv ,proj,projproj 11
)projproj(
,proj,proj 11
vvv
uuvvuuvv
WW
ttWW
60. Ex 5: (Projection onto a subspace)
3,1,1,0,0,2,1,3,0 21 vww
Find the projection of the vector v onto the subspace W.
:, 21 ww
Sol:
an orthogonal basis for W
:0,0,1),
10
1
,
10
3
,0(,,
2
2
1
1
21
w
w
w
w
uu
an orthonormal basis for W
}),({ 21 wwspanW
)
5
3
,
5
9
,1(0,0,1)
10
1
,
10
3
,0(
10
6
,,proj 2211
uuvuuvvW
61. Find by the other method:
bb
b
b
vbww
T1T
)(
T1T
21
)(proj
)(
,,
AAAAAx
AAAx
Ax
A
Acs
62. • Thm 5.15: (Orthogonal projection and distance)
Let W be a subspace of an inner product space V,
and .
Then for all ,
Vv
Ww vw Wproj
||||||proj|| wvvv W
||||min||proj||ro W
wvvv wW
( is the best approximation to v from W)vWproj
63. Pf:
)proj()proj( wvvvwv WW
)proj()proj( wvvv WW
By the Pythagorean theorem
222
||proj||||proj|||||| wvvvwv WW
0projproj wvvw WW
22
||proj|||||| vvwv W
||||||proj|| wvvv W
64. • Notes:
(1) Among all the scalar multiples of a vector u,
the
orthogonal projection of v onto u is the one
that is
closest to v. (p.302 Thm 5.9)
(2) Among all the vectors in the subspace W, the
vector
is the closest vector to v.
vWproj
65. • Thm 5.16: (Fundamental subspaces of a matrix)
If A is an m×n matrix, then
(1)
(2)
(3)
(4)
)())((
)())((
ACSANS
ANSACS
)())((
)())((
ACSANS
ANSACS
mmT
RANSACSRANSACS ))(()()()(
nTnT
RACSACSRANSACS ))(()()()(
66. Ex 6: (Fundamental subspaces)
Find the four fundamental subspaces of the
matrix.
000
000
100
021
A (reduced row-echelon form)
Sol:
4
ofsubspaceais0,0,1,00,0,0,1span)( RACS
3
ofsubspaceais1,0,00,2,1span)( RARSACS
3
ofsubspaceais0,1,2span)( RANS
68. • Ex 3 & Ex 4:
Let W is a subspace of R4
and .
(a) Find a basis for W
(b) Find a basis for the orthogonal complement of W.
)10,0,0,(),01,2,1,( 21 ww
Sol:
21
00
00
10
01
~
10
01
02
01
ww
RA (reduced row-echelon form)
}),({ 21 wwspanW
69. 1,0,0,0,0,1,2,1
)( ACSWa
is a basis for W
W
ts
t
s
ts
x
x
x
x
A
ANSACSWb
forbasisais0,1,0,10,0,1,2
0
1
0
1
0
0
1
2
0
2
1000
0121
)(
4
3
2
1
• Notes:
4
4
(2)
)dim()dim()dim((1)
RWW
RWW
70. Least squares problem:
(A system of linear equations)
(1) When the system is consistent, we can use the Gaussian
elimination with back-substitution to solve for x
bxA
11 mnnm
(2) When the system is inconsistent, how to find the “best possible”
solution of the system. That is, the value of x for which the difference
between Ax and b is small.
73. Note: (Ax = b is an inconsistent system)
The problem of finding the least squares solution of
is equal to he problem of finding an exact solution of
the associated normal system .
bxA
bx AAA ˆ
74. • Ex 7: (Solving the normal equations)
Find the least squares solution of the
following system
(this system is inconsistent)
and find the orthogonal projection of b on the column
space of A.
3
1
0
31
21
11
1
0
c
c
A bx
76. the least squares solution of Ax = b
2
3
3
5
ˆx
the orthogonal projection of b on the column space of A
6
17
6
8
6
1
2
3
3
5
)(
31
21
11
ˆproj xb AACS