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.
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.
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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GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
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The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
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Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
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Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
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A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
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Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
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During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
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See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
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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