SlideShare a Scribd company logo
In P_2 is S_2 = {x^2 + 1, x^2 + x, x + 6} a linearly independent set? Justify your answer! Does
S_2 span P_2 ? Justify your answer! Is S_2 a basis for P_2? (clearly state why or why not)
Solution
lets calculate the dimension of S_2
It is clear S2 is a subspace of P2 since every element of S2 is in P2.
So if the dimension of S2 is 3 i.e. all elements of S2 are linearly independent then S2 spans P2
and it is a basis for P2
otherwise Not.
Now lets see if all elements are LI
if c1 (x^2 +1) + c2(x^2 + x) + c3(x+6) = 0 (zero polynomial)
then
c1 = -c2
c1 = -6c3
c2 = -c3
which gives all c1,c2,c3 = 0
so they are linearly independent and hence its a basis for P2.

More Related Content

More from arcotstarsports

The principal role of ATP In the cell is A) serving as structural s.pdf
The principal role of ATP In the cell is  A) serving as structural s.pdfThe principal role of ATP In the cell is  A) serving as structural s.pdf
The principal role of ATP In the cell is A) serving as structural s.pdf
arcotstarsports
 
The neutral theory of evolution proposed that most genetic mutations .pdf
The neutral theory of evolution proposed that most genetic mutations .pdfThe neutral theory of evolution proposed that most genetic mutations .pdf
The neutral theory of evolution proposed that most genetic mutations .pdf
arcotstarsports
 
The oldest living organisms on Earth are plants. Some bristlecone pi.pdf
The oldest living organisms on Earth are plants. Some bristlecone pi.pdfThe oldest living organisms on Earth are plants. Some bristlecone pi.pdf
The oldest living organisms on Earth are plants. Some bristlecone pi.pdf
arcotstarsports
 
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdfSuppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
arcotstarsports
 
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdfResearch the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
arcotstarsports
 
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdf
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdfSpecies I Species II Species III Species IV Ancestral sequence IV Sit.pdf
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdf
arcotstarsports
 
Solve the differential equation y+y=sin(x). Im thinking I ne.pdf
Solve the differential equation y+y=sin(x). Im thinking I ne.pdfSolve the differential equation y+y=sin(x). Im thinking I ne.pdf
Solve the differential equation y+y=sin(x). Im thinking I ne.pdf
arcotstarsports
 
Please write a non-recursive procedure to print all nodes for a bina.pdf
Please write a non-recursive procedure to print all nodes for a bina.pdfPlease write a non-recursive procedure to print all nodes for a bina.pdf
Please write a non-recursive procedure to print all nodes for a bina.pdf
arcotstarsports
 
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdfPlease do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
arcotstarsports
 
need help with code I wrote. This code is a maze gui, and i need hel.pdf
need help with code I wrote. This code is a maze gui, and i need hel.pdfneed help with code I wrote. This code is a maze gui, and i need hel.pdf
need help with code I wrote. This code is a maze gui, and i need hel.pdf
arcotstarsports
 
Microbiology question Which of the following is the most unique a.pdf
Microbiology question Which of the following is the most unique a.pdfMicrobiology question Which of the following is the most unique a.pdf
Microbiology question Which of the following is the most unique a.pdf
arcotstarsports
 
Is Event B dependent or independent of Event A A green ball is draw.pdf
Is Event B dependent or independent of Event A A green ball is draw.pdfIs Event B dependent or independent of Event A A green ball is draw.pdf
Is Event B dependent or independent of Event A A green ball is draw.pdf
arcotstarsports
 
In this view, the front of the heart has been reflected upwards. Ide.pdf
In this view, the front of the heart has been reflected upwards. Ide.pdfIn this view, the front of the heart has been reflected upwards. Ide.pdf
In this view, the front of the heart has been reflected upwards. Ide.pdf
arcotstarsports
 
I need to find the area of the shaded region. I dont have the grap.pdf
I need to find the area of the shaded region. I dont have the grap.pdfI need to find the area of the shaded region. I dont have the grap.pdf
I need to find the area of the shaded region. I dont have the grap.pdf
arcotstarsports
 
For Programming Embedded SystemsQ-02 Which structure variable poi.pdf
For Programming Embedded SystemsQ-02 Which structure variable poi.pdfFor Programming Embedded SystemsQ-02 Which structure variable poi.pdf
For Programming Embedded SystemsQ-02 Which structure variable poi.pdf
arcotstarsports
 
Florida State stat club is sending a delegate of 2 members to attend.pdf
Florida State stat club is sending a delegate of 2 members to attend.pdfFlorida State stat club is sending a delegate of 2 members to attend.pdf
Florida State stat club is sending a delegate of 2 members to attend.pdf
arcotstarsports
 
Fill in the blank to make a true statement. Natural numbers that are.pdf
Fill in the blank to make a true statement.  Natural numbers that are.pdfFill in the blank to make a true statement.  Natural numbers that are.pdf
Fill in the blank to make a true statement. Natural numbers that are.pdf
arcotstarsports
 
Define metabolism and name the two phases of metabolism. Briefly.pdf
Define metabolism and name the two phases of metabolism.  Briefly.pdfDefine metabolism and name the two phases of metabolism.  Briefly.pdf
Define metabolism and name the two phases of metabolism. Briefly.pdf
arcotstarsports
 
Evaluate the limit using LHospitals rule Suppose that S.pdf
Evaluate the limit using LHospitals rule  Suppose that  S.pdfEvaluate the limit using LHospitals rule  Suppose that  S.pdf
Evaluate the limit using LHospitals rule Suppose that S.pdf
arcotstarsports
 
Discuss at least three elements of power and specific strategies to .pdf
Discuss at least three elements of power and specific strategies to .pdfDiscuss at least three elements of power and specific strategies to .pdf
Discuss at least three elements of power and specific strategies to .pdf
arcotstarsports
 

More from arcotstarsports (20)

The principal role of ATP In the cell is A) serving as structural s.pdf
The principal role of ATP In the cell is  A) serving as structural s.pdfThe principal role of ATP In the cell is  A) serving as structural s.pdf
The principal role of ATP In the cell is A) serving as structural s.pdf
 
The neutral theory of evolution proposed that most genetic mutations .pdf
The neutral theory of evolution proposed that most genetic mutations .pdfThe neutral theory of evolution proposed that most genetic mutations .pdf
The neutral theory of evolution proposed that most genetic mutations .pdf
 
The oldest living organisms on Earth are plants. Some bristlecone pi.pdf
The oldest living organisms on Earth are plants. Some bristlecone pi.pdfThe oldest living organisms on Earth are plants. Some bristlecone pi.pdf
The oldest living organisms on Earth are plants. Some bristlecone pi.pdf
 
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdfSuppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
Suppose there is a person whose total lung capacity (TLC) is 4.90 L, .pdf
 
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdfResearch the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
Research the IPv4 ns the IPv6 protocols, then prepare a report that .pdf
 
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdf
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdfSpecies I Species II Species III Species IV Ancestral sequence IV Sit.pdf
Species I Species II Species III Species IV Ancestral sequence IV Sit.pdf
 
Solve the differential equation y+y=sin(x). Im thinking I ne.pdf
Solve the differential equation y+y=sin(x). Im thinking I ne.pdfSolve the differential equation y+y=sin(x). Im thinking I ne.pdf
Solve the differential equation y+y=sin(x). Im thinking I ne.pdf
 
Please write a non-recursive procedure to print all nodes for a bina.pdf
Please write a non-recursive procedure to print all nodes for a bina.pdfPlease write a non-recursive procedure to print all nodes for a bina.pdf
Please write a non-recursive procedure to print all nodes for a bina.pdf
 
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdfPlease do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
Please do C, D, and E For the given f Z right arrow Z, decide wheth.pdf
 
need help with code I wrote. This code is a maze gui, and i need hel.pdf
need help with code I wrote. This code is a maze gui, and i need hel.pdfneed help with code I wrote. This code is a maze gui, and i need hel.pdf
need help with code I wrote. This code is a maze gui, and i need hel.pdf
 
Microbiology question Which of the following is the most unique a.pdf
Microbiology question Which of the following is the most unique a.pdfMicrobiology question Which of the following is the most unique a.pdf
Microbiology question Which of the following is the most unique a.pdf
 
Is Event B dependent or independent of Event A A green ball is draw.pdf
Is Event B dependent or independent of Event A A green ball is draw.pdfIs Event B dependent or independent of Event A A green ball is draw.pdf
Is Event B dependent or independent of Event A A green ball is draw.pdf
 
In this view, the front of the heart has been reflected upwards. Ide.pdf
In this view, the front of the heart has been reflected upwards. Ide.pdfIn this view, the front of the heart has been reflected upwards. Ide.pdf
In this view, the front of the heart has been reflected upwards. Ide.pdf
 
I need to find the area of the shaded region. I dont have the grap.pdf
I need to find the area of the shaded region. I dont have the grap.pdfI need to find the area of the shaded region. I dont have the grap.pdf
I need to find the area of the shaded region. I dont have the grap.pdf
 
For Programming Embedded SystemsQ-02 Which structure variable poi.pdf
For Programming Embedded SystemsQ-02 Which structure variable poi.pdfFor Programming Embedded SystemsQ-02 Which structure variable poi.pdf
For Programming Embedded SystemsQ-02 Which structure variable poi.pdf
 
Florida State stat club is sending a delegate of 2 members to attend.pdf
Florida State stat club is sending a delegate of 2 members to attend.pdfFlorida State stat club is sending a delegate of 2 members to attend.pdf
Florida State stat club is sending a delegate of 2 members to attend.pdf
 
Fill in the blank to make a true statement. Natural numbers that are.pdf
Fill in the blank to make a true statement.  Natural numbers that are.pdfFill in the blank to make a true statement.  Natural numbers that are.pdf
Fill in the blank to make a true statement. Natural numbers that are.pdf
 
Define metabolism and name the two phases of metabolism. Briefly.pdf
Define metabolism and name the two phases of metabolism.  Briefly.pdfDefine metabolism and name the two phases of metabolism.  Briefly.pdf
Define metabolism and name the two phases of metabolism. Briefly.pdf
 
Evaluate the limit using LHospitals rule Suppose that S.pdf
Evaluate the limit using LHospitals rule  Suppose that  S.pdfEvaluate the limit using LHospitals rule  Suppose that  S.pdf
Evaluate the limit using LHospitals rule Suppose that S.pdf
 
Discuss at least three elements of power and specific strategies to .pdf
Discuss at least three elements of power and specific strategies to .pdfDiscuss at least three elements of power and specific strategies to .pdf
Discuss at least three elements of power and specific strategies to .pdf
 

Recently uploaded

Language Across the Curriculm LAC B.Ed.
Language Across the  Curriculm LAC B.Ed.Language Across the  Curriculm LAC B.Ed.
Language Across the Curriculm LAC B.Ed.
Atul Kumar Singh
 
How to Make a Field invisible in Odoo 17
How to Make a Field invisible in Odoo 17How to Make a Field invisible in Odoo 17
How to Make a Field invisible in Odoo 17
Celine George
 
Honest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptxHonest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptx
timhan337
 
Unit 8 - Information and Communication Technology (Paper I).pdf
Unit 8 - Information and Communication Technology (Paper I).pdfUnit 8 - Information and Communication Technology (Paper I).pdf
Unit 8 - Information and Communication Technology (Paper I).pdf
Thiyagu K
 
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCECLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
BhavyaRajput3
 
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdfAdversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Po-Chuan Chen
 
special B.ed 2nd year old paper_20240531.pdf
special B.ed 2nd year old paper_20240531.pdfspecial B.ed 2nd year old paper_20240531.pdf
special B.ed 2nd year old paper_20240531.pdf
Special education needs
 
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
Welcome to TechSoup   New Member Orientation and Q&A (May 2024).pdfWelcome to TechSoup   New Member Orientation and Q&A (May 2024).pdf
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
TechSoup
 
A Strategic Approach: GenAI in Education
A Strategic Approach: GenAI in EducationA Strategic Approach: GenAI in Education
A Strategic Approach: GenAI in Education
Peter Windle
 
Instructions for Submissions thorugh G- Classroom.pptx
Instructions for Submissions thorugh G- Classroom.pptxInstructions for Submissions thorugh G- Classroom.pptx
Instructions for Submissions thorugh G- Classroom.pptx
Jheel Barad
 
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
MysoreMuleSoftMeetup
 
The approach at University of Liverpool.pptx
The approach at University of Liverpool.pptxThe approach at University of Liverpool.pptx
The approach at University of Liverpool.pptx
Jisc
 
Thesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.pptThesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.ppt
EverAndrsGuerraGuerr
 
Synthetic Fiber Construction in lab .pptx
Synthetic Fiber Construction in lab .pptxSynthetic Fiber Construction in lab .pptx
Synthetic Fiber Construction in lab .pptx
Pavel ( NSTU)
 
The basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptxThe basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptx
heathfieldcps1
 
Home assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdfHome assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdf
Tamralipta Mahavidyalaya
 
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
siemaillard
 
Chapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptxChapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptx
Mohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
 
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
Nguyen Thanh Tu Collection
 
How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...
Jisc
 

Recently uploaded (20)

Language Across the Curriculm LAC B.Ed.
Language Across the  Curriculm LAC B.Ed.Language Across the  Curriculm LAC B.Ed.
Language Across the Curriculm LAC B.Ed.
 
How to Make a Field invisible in Odoo 17
How to Make a Field invisible in Odoo 17How to Make a Field invisible in Odoo 17
How to Make a Field invisible in Odoo 17
 
Honest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptxHonest Reviews of Tim Han LMA Course Program.pptx
Honest Reviews of Tim Han LMA Course Program.pptx
 
Unit 8 - Information and Communication Technology (Paper I).pdf
Unit 8 - Information and Communication Technology (Paper I).pdfUnit 8 - Information and Communication Technology (Paper I).pdf
Unit 8 - Information and Communication Technology (Paper I).pdf
 
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCECLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
 
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdfAdversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
Adversarial Attention Modeling for Multi-dimensional Emotion Regression.pdf
 
special B.ed 2nd year old paper_20240531.pdf
special B.ed 2nd year old paper_20240531.pdfspecial B.ed 2nd year old paper_20240531.pdf
special B.ed 2nd year old paper_20240531.pdf
 
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
Welcome to TechSoup   New Member Orientation and Q&A (May 2024).pdfWelcome to TechSoup   New Member Orientation and Q&A (May 2024).pdf
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
 
A Strategic Approach: GenAI in Education
A Strategic Approach: GenAI in EducationA Strategic Approach: GenAI in Education
A Strategic Approach: GenAI in Education
 
Instructions for Submissions thorugh G- Classroom.pptx
Instructions for Submissions thorugh G- Classroom.pptxInstructions for Submissions thorugh G- Classroom.pptx
Instructions for Submissions thorugh G- Classroom.pptx
 
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
Mule 4.6 & Java 17 Upgrade | MuleSoft Mysore Meetup #46
 
The approach at University of Liverpool.pptx
The approach at University of Liverpool.pptxThe approach at University of Liverpool.pptx
The approach at University of Liverpool.pptx
 
Thesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.pptThesis Statement for students diagnonsed withADHD.ppt
Thesis Statement for students diagnonsed withADHD.ppt
 
Synthetic Fiber Construction in lab .pptx
Synthetic Fiber Construction in lab .pptxSynthetic Fiber Construction in lab .pptx
Synthetic Fiber Construction in lab .pptx
 
The basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptxThe basics of sentences session 5pptx.pptx
The basics of sentences session 5pptx.pptx
 
Home assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdfHome assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdf
 
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
 
Chapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptxChapter 3 - Islamic Banking Products and Services.pptx
Chapter 3 - Islamic Banking Products and Services.pptx
 
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
 
How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...How libraries can support authors with open access requirements for UKRI fund...
How libraries can support authors with open access requirements for UKRI fund...
 

In P_2 is S_2 = {x^2 + 1, x^2 + x, x + 6} a linearly independent set.pdf

  • 1. In P_2 is S_2 = {x^2 + 1, x^2 + x, x + 6} a linearly independent set? Justify your answer! Does S_2 span P_2 ? Justify your answer! Is S_2 a basis for P_2? (clearly state why or why not) Solution lets calculate the dimension of S_2 It is clear S2 is a subspace of P2 since every element of S2 is in P2. So if the dimension of S2 is 3 i.e. all elements of S2 are linearly independent then S2 spans P2 and it is a basis for P2 otherwise Not. Now lets see if all elements are LI if c1 (x^2 +1) + c2(x^2 + x) + c3(x+6) = 0 (zero polynomial) then c1 = -c2 c1 = -6c3 c2 = -c3 which gives all c1,c2,c3 = 0 so they are linearly independent and hence its a basis for P2.