Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D....
Transcript of Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D....
![Page 1: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/1.jpg)
Linear IndependenceMath 218
Brian D. Fitzpatrick
Duke University
March 3, 2020
MATH
![Page 2: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/2.jpg)
Overview
Geometric MotivationCounting “Directions”
BackgroundLinear Combinations as Matrix MultiplicationAn Important Observation
Definitions and ExamplesDefinitionsExamples
The Linear Independence TestStatementExample
The “Pivot Columns” of a MatrixDefinitionExample
![Page 3: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/3.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 4: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/4.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 5: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/5.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 6: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/6.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v
3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 7: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/7.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 8: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/8.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v
3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 9: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/9.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 10: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/10.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v
−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 11: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/11.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 12: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/12.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v
−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 13: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/13.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 14: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/14.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 15: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/15.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v } “look like”?
AnswerSpan{ #»v } consists of all “multiples” of { #»v }.
Span{ #»v }
2 · #»v3 · #»v
−2 · #»v−3 · #»v
#»v
Assuming #»v 6= #»
O , Span{ #»v } is the line containing #»v .
![Page 16: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/16.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 17: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/17.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 18: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/18.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 19: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/19.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 20: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/20.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 21: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/21.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 22: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/22.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 23: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/23.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 24: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/24.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 25: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/25.jpg)
Geometric MotivationCounting “Directions”
QuestionWhat does Span{ #»v 1,
#»v 2} “look like”?
AnswerSpan{ #»v 1,
#»v 2} consists of all “linear combinations” of { #»v 1,#»v 2}.
Span{#»v 1,#»v 2}
#»v 1
#»v 2
Assuming #»v 1 and #»v 2 are not parallel, Span{ #»v 1,#»v 2} is the plane
containing #»v 1 and #»v 2.
![Page 26: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/26.jpg)
Geometric MotivationCounting “Directions”
Span{ #»v }Defines a one-directional object (line) if #»v 6= #»
O .
Span{ #»v 1,#»v 2}
Defines a two-directional object (plane) if #»v 1 and #»v 2 are notparallel.
QuestionHow can we determine if Span{ #»v 1,
#»v 2, . . . ,#»v n} defines an
n-directional object?
AnswerSpan{ #»v 1,
#»v 2, . . . ,#»v n} defines an n-directional object if the list
{ #»v 1,#»v 2, . . . ,
#»v n} is linearly independent.
![Page 27: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/27.jpg)
Geometric MotivationCounting “Directions”
Span{ #»v }Defines a one-directional object (line) if #»v 6= #»
O .
Span{ #»v 1,#»v 2}
Defines a two-directional object (plane) if #»v 1 and #»v 2 are notparallel.
QuestionHow can we determine if Span{ #»v 1,
#»v 2, . . . ,#»v n} defines an
n-directional object?
AnswerSpan{ #»v 1,
#»v 2, . . . ,#»v n} defines an n-directional object if the list
{ #»v 1,#»v 2, . . . ,
#»v n} is linearly independent.
![Page 28: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/28.jpg)
Geometric MotivationCounting “Directions”
Span{ #»v }Defines a one-directional object (line) if #»v 6= #»
O .
Span{ #»v 1,#»v 2}
Defines a two-directional object (plane) if #»v 1 and #»v 2 are notparallel.
QuestionHow can we determine if Span{ #»v 1,
#»v 2, . . . ,#»v n} defines an
n-directional object?
AnswerSpan{ #»v 1,
#»v 2, . . . ,#»v n} defines an n-directional object if the list
{ #»v 1,#»v 2, . . . ,
#»v n} is linearly independent.
![Page 29: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/29.jpg)
Geometric MotivationCounting “Directions”
Span{ #»v }Defines a one-directional object (line) if #»v 6= #»
O .
Span{ #»v 1,#»v 2}
Defines a two-directional object (plane) if #»v 1 and #»v 2 are notparallel.
QuestionHow can we determine if Span{ #»v 1,
#»v 2, . . . ,#»v n} defines an
n-directional object?
AnswerSpan{ #»v 1,
#»v 2, . . . ,#»v n} defines an n-directional object if the list
{ #»v 1,#»v 2, . . . ,
#»v n} is linearly independent.
![Page 30: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/30.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 31: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/31.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3.
Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 32: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/32.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 33: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/33.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)
#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 34: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/34.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)
#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 35: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/35.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)
#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 36: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/36.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)
#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 37: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/37.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 38: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/38.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
−5 −15 −10 −20−2 −6 −4 −8
2 6 4 8
Note that Col(A) ⊂ R3. Each column is a multiple of the firstcolumn.
3 · #»a 1
−4 · #»a 1
−2 · #»a 1
Col(A)#»a 1
This illustrates that Col(A) = Span{〈−5, −2, 2〉 }.
![Page 39: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/39.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 40: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/40.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 41: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/41.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3
#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 42: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/42.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3
#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 43: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/43.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3
#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 44: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/44.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 45: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/45.jpg)
Geometric MotivationCounting “Directions”
Example
Consider the matrix A given by
A =
1 2 7 6−8 −16 3 11
3 6 −2 −52 4 −9 −11
Note that Col(A) ⊂ R4.
2 · #»a 1
#»a 3
− #»a 1 + #»a 3#»a 1
This means that Col(A) = Span{〈1, −8, 3, 2〉 , 〈7, 3, −2, −9〉 }.
![Page 46: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/46.jpg)
BackgroundLinear Combinations as Matrix Multiplication
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm.
Every linearcombination
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
b
is of the form A #»c =#»
b where
A =[
#»v 1#»v 2 · · · #»v n
]#»c =
c1c2...cn
#»
b =
b1b2...bm
Note that A is an m × n matrix.
![Page 47: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/47.jpg)
BackgroundLinear Combinations as Matrix Multiplication
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. Every linear
combination
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
b
is of the form A #»c =#»
b where
A =[
#»v 1#»v 2 · · · #»v n
]#»c =
c1c2...cn
#»
b =
b1b2...bm
Note that A is an m × n matrix.
![Page 48: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/48.jpg)
BackgroundLinear Combinations as Matrix Multiplication
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. Every linear
combination
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
b
is of the form A #»c =#»
b where
A =[
#»v 1#»v 2 · · · #»v n
]#»c =
c1c2...cn
#»
b =
b1b2...bm
Note that A is an m × n matrix.
![Page 49: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/49.jpg)
BackgroundLinear Combinations as Matrix Multiplication
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. Every linear
combination
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
b
is of the form A #»c =#»
b where
A =[
#»v 1#»v 2 · · · #»v n
]#»c =
c1c2...cn
#»
b =
b1b2...bm
Note that A is an m × n matrix.
![Page 50: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/50.jpg)
BackgroundLinear Combinations as Matrix Multiplication
Example
The linear combination
c1 ·[
131
]+ c2 ·
[0−3
]+ c3 ·
[7−5
]+ c4 ·
[−5−3
]=
[33−11
]may be written as
[1 0 7 −5
31 −3 −5 −3
] c1c2c3c4
=
[33−11
]
![Page 51: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/51.jpg)
BackgroundLinear Combinations as Matrix Multiplication
Example
The linear combination
c1 ·[
131
]+ c2 ·
[0−3
]+ c3 ·
[7−5
]+ c4 ·
[−5−3
]=
[33−11
]may be written as
[1 0 7 −5
31 −3 −5 −3
] c1c2c3c4
=
[33−11
]
![Page 52: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/52.jpg)
BackgroundAn Important Observation
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. The equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O (∗)
can always be solved by c1 = c2 = · · · = cn = 0.
This is called thetrivial linear combination.
QuestionWhen is (∗) solved by a nontrivial linear combination?
![Page 53: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/53.jpg)
BackgroundAn Important Observation
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. The equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O (∗)
can always be solved by c1 = c2 = · · · = cn = 0. This is called thetrivial linear combination.
QuestionWhen is (∗) solved by a nontrivial linear combination?
![Page 54: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/54.jpg)
BackgroundAn Important Observation
ObservationLet { #»v 1,
#»v 2, . . . ,#»v n} be a list of vectors in Rm. The equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O (∗)
can always be solved by c1 = c2 = · · · = cn = 0. This is called thetrivial linear combination.
QuestionWhen is (∗) solved by a nontrivial linear combination?
![Page 55: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/55.jpg)
Definitions and ExamplesDefinitions
DefinitionA list of vectors { #»v 1,
#»v 2, . . . ,#»v n} in Rm is linearly independent if
the only solution to
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
is the trivial solution c1 = c2 = · · · = cn = 0.
The list{ #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if it is not linearly
independent.
NoteThe list { #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if the equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
has a nontrivial solution.
![Page 56: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/56.jpg)
Definitions and ExamplesDefinitions
DefinitionA list of vectors { #»v 1,
#»v 2, . . . ,#»v n} in Rm is linearly independent if
the only solution to
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
is the trivial solution c1 = c2 = · · · = cn = 0. The list{ #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if it is not linearly
independent.
NoteThe list { #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if the equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
has a nontrivial solution.
![Page 57: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/57.jpg)
Definitions and ExamplesDefinitions
DefinitionA list of vectors { #»v 1,
#»v 2, . . . ,#»v n} in Rm is linearly independent if
the only solution to
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
is the trivial solution c1 = c2 = · · · = cn = 0. The list{ #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if it is not linearly
independent.
NoteThe list { #»v 1,
#»v 2, . . . ,#»v n} is linearly dependent if the equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ cn · #»v n =#»
O
has a nontrivial solution.
![Page 58: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/58.jpg)
Definitions and ExamplesExamples
Example
Note that
(3)
10−4
+ (1)
−21
15
+ (1)
−1−1−3
+ (0)
−53
41
=
000
This means that columns of 1 −2 −1 −50 1 −1 3−4 15 −3 41
are linearly dependent.
![Page 59: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/59.jpg)
Definitions and ExamplesExamples
Example
Note that
(3)
10−4
+ (1)
−21
15
+ (1)
−1−1−3
+ (0)
−53
41
=
000
This means that columns of 1 −2 −1 −5
0 1 −1 3−4 15 −3 41
are linearly dependent.
![Page 60: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/60.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0. Hence { #»v 1,
#»v 2,#»v 3} is linearly
independent.
![Page 61: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/61.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0. Hence { #»v 1,
#»v 2,#»v 3} is linearly
independent.
![Page 62: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/62.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0. Hence { #»v 1,
#»v 2,#»v 3} is linearly
independent.
![Page 63: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/63.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0. Hence { #»v 1,#»v 2,
#»v 3} is linearlyindependent.
![Page 64: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/64.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0.
Hence { #»v 1,#»v 2,
#»v 3} is linearlyindependent.
![Page 65: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/65.jpg)
Definitions and ExamplesExamples
Example
Determine if the list { #»v 1,#»v 2,
#»v 3} is linearly independent where
#»v 1 = 〈−1, 1, 1, 2〉 #»v 2 = 〈1, −2, −1, −3〉 #»v 3 = 〈1, 0, 0, 2〉
To determine if { #»v 1,#»v 2,
#»v 3} is linearly independent, we consider
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 =#»
O
This gives the system−1 1 1 0
1 −2 0 01 −1 0 02 −3 2 0
1 0 0 00 1 0 00 0 1 00 0 0 0
Thus c1 = c2 = c3 = 0. Hence { #»v 1,
#»v 2,#»v 3} is linearly
independent.
![Page 66: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/66.jpg)
Definitions and ExamplesExamples
Example
Suppose that { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent. Show that
{ #»v 1 − #»v 4,#»v 2 − #»v 4,
#»v 3 − #»v 4} is linearly independent.
SolutionSuppose that
c1 · ( #»v 1 − #»v 4) + c2 · ( #»v 2 − #»v 4) + c3 · ( #»v 3 − #»v 4) =#»
O
Then
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 + (−c1 − c2 − c3) · #»v 4 =#»
O (∗)
Since { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent, each coefficient in
(∗) must be zero. In particular, c1 = c2 = c3 = 0.
![Page 67: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/67.jpg)
Definitions and ExamplesExamples
Example
Suppose that { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent. Show that
{ #»v 1 − #»v 4,#»v 2 − #»v 4,
#»v 3 − #»v 4} is linearly independent.
SolutionSuppose that
c1 · ( #»v 1 − #»v 4) + c2 · ( #»v 2 − #»v 4) + c3 · ( #»v 3 − #»v 4) =#»
O
Then
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 + (−c1 − c2 − c3) · #»v 4 =#»
O (∗)
Since { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent, each coefficient in
(∗) must be zero. In particular, c1 = c2 = c3 = 0.
![Page 68: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/68.jpg)
Definitions and ExamplesExamples
Example
Suppose that { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent. Show that
{ #»v 1 − #»v 4,#»v 2 − #»v 4,
#»v 3 − #»v 4} is linearly independent.
SolutionSuppose that
c1 · ( #»v 1 − #»v 4) + c2 · ( #»v 2 − #»v 4) + c3 · ( #»v 3 − #»v 4) =#»
O
Then
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 + (−c1 − c2 − c3) · #»v 4 =#»
O (∗)
Since { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent, each coefficient in
(∗) must be zero. In particular, c1 = c2 = c3 = 0.
![Page 69: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/69.jpg)
Definitions and ExamplesExamples
Example
Suppose that { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent. Show that
{ #»v 1 − #»v 4,#»v 2 − #»v 4,
#»v 3 − #»v 4} is linearly independent.
SolutionSuppose that
c1 · ( #»v 1 − #»v 4) + c2 · ( #»v 2 − #»v 4) + c3 · ( #»v 3 − #»v 4) =#»
O
Then
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 + (−c1 − c2 − c3) · #»v 4 =#»
O (∗)
Since { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent, each coefficient in
(∗) must be zero.
In particular, c1 = c2 = c3 = 0.
![Page 70: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/70.jpg)
Definitions and ExamplesExamples
Example
Suppose that { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent. Show that
{ #»v 1 − #»v 4,#»v 2 − #»v 4,
#»v 3 − #»v 4} is linearly independent.
SolutionSuppose that
c1 · ( #»v 1 − #»v 4) + c2 · ( #»v 2 − #»v 4) + c3 · ( #»v 3 − #»v 4) =#»
O
Then
c1 · #»v 1 + c2 · #»v 2 + c3 · #»v 3 + (−c1 − c2 − c3) · #»v 4 =#»
O (∗)
Since { #»v 1,#»v 2,
#»v 3,#»v 4} is linearly independent, each coefficient in
(∗) must be zero. In particular, c1 = c2 = c3 = 0.
![Page 71: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/71.jpg)
The Linear Independence TestStatement
NoteThe equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ ck · #»v k =#»
O
is given by the augmented matrix[#»v 1
#»v 2 · · · #»v k#»
O]
So, { #»v 1,#»v 2, . . . ,
#»v k} is linearly independent if and only if
A =[
#»v 1#»v 2 · · · #»v k
]has full column rank.
![Page 72: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/72.jpg)
The Linear Independence TestStatement
NoteThe equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ ck · #»v k =#»
O
is given by the augmented matrix[#»v 1
#»v 2 · · · #»v k#»
O]
So, { #»v 1,#»v 2, . . . ,
#»v k} is linearly independent if and only if
A =[
#»v 1#»v 2 · · · #»v k
]has
full column rank.
![Page 73: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/73.jpg)
The Linear Independence TestStatement
NoteThe equation
c1 · #»v 1 + c2 · #»v 2 + · · ·+ ck · #»v k =#»
O
is given by the augmented matrix[#»v 1
#»v 2 · · · #»v k#»
O]
So, { #»v 1,#»v 2, . . . ,
#»v k} is linearly independent if and only if
A =[
#»v 1#»v 2 · · · #»v k
]has full column rank.
![Page 74: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/74.jpg)
The Linear Independence TestStatement
Theorem (The Linear Independence Test)
The list { #»v 1,#»v 2, . . . ,
#»v k} is linearly independent if and only if[#»v 1
#»v 2 · · · #»v k
]has full column rank.
![Page 75: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/75.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−14, −5, −5, −3〉 , 〈−53, −19, −18, −13〉 , 〈78, 28, 26, 20〉 }
is linearly independent.
SolutionNote that
rref
−14 −53 78−5 −19 28−5 −18 26−3 −13 20
=
1 0 20 1 −20 0 00 0 0
Since rank = 2 < # columns, the list is linearly dependent.
![Page 76: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/76.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−14, −5, −5, −3〉 , 〈−53, −19, −18, −13〉 , 〈78, 28, 26, 20〉 }
is linearly independent.
SolutionNote that
rref
−14 −53 78−5 −19 28−5 −18 26−3 −13 20
=
1 0 20 1 −20 0 00 0 0
Since rank = 2 < # columns, the list is linearly dependent.
![Page 77: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/77.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−14, −5, −5, −3〉 , 〈−53, −19, −18, −13〉 , 〈78, 28, 26, 20〉 }
is linearly independent.
SolutionNote that
rref
−14 −53 78−5 −19 28−5 −18 26−3 −13 20
=
1 0 20 1 −20 0 00 0 0
Since rank = 2 < # columns, the list is linearly dependent.
![Page 78: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/78.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−1, −5, −1, 1〉 , 〈7, −14, −1, −2〉 , 〈33, −31, 2, −13〉 }
is linearly independent.
SolutionNote that
rref
−1 7 33−5 −14 −31−1 −1 2
1 −2 −13
=
1 0 00 1 00 0 10 0 0
Since rank = 3 = # columns, the list is linearly independent.
![Page 79: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/79.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−1, −5, −1, 1〉 , 〈7, −14, −1, −2〉 , 〈33, −31, 2, −13〉 }
is linearly independent.
SolutionNote that
rref
−1 7 33−5 −14 −31−1 −1 2
1 −2 −13
=
1 0 00 1 00 0 10 0 0
Since rank = 3 = # columns, the list is linearly independent.
![Page 80: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/80.jpg)
The Linear Independence TestExample
Example
Determine if the list
{〈−1, −5, −1, 1〉 , 〈7, −14, −1, −2〉 , 〈33, −31, 2, −13〉 }
is linearly independent.
SolutionNote that
rref
−1 7 33−5 −14 −31−1 −1 2
1 −2 −13
=
1 0 00 1 00 0 10 0 0
Since rank = 3 = # columns, the list is linearly independent.
![Page 81: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/81.jpg)
The Linear Independence TestExample
Example
Determine if the list 1
5−5
,
−2−914
,
−6−29
35
,
526−23
is linearly independent.
SolutionThe matrix
A =
1 −2 −6 55 −9 −29 26−5 14 35 −23
satisfies rank(A) ≤ 3. Since rank(A) 6= # columns = 4, the list isnot linearly independent.
![Page 82: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/82.jpg)
The Linear Independence TestExample
Example
Determine if the list 1
5−5
,
−2−914
,
−6−29
35
,
526−23
is linearly independent.
SolutionThe matrix
A =
1 −2 −6 55 −9 −29 26−5 14 35 −23
satisfies rank(A) ≤
3. Since rank(A) 6= # columns = 4, the list isnot linearly independent.
![Page 83: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/83.jpg)
The Linear Independence TestExample
Example
Determine if the list 1
5−5
,
−2−914
,
−6−29
35
,
526−23
is linearly independent.
SolutionThe matrix
A =
1 −2 −6 55 −9 −29 26−5 14 35 −23
satisfies rank(A) ≤ 3.
Since rank(A) 6= # columns = 4, the list isnot linearly independent.
![Page 84: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/84.jpg)
The Linear Independence TestExample
Example
Determine if the list 1
5−5
,
−2−914
,
−6−29
35
,
526−23
is linearly independent.
SolutionThe matrix
A =
1 −2 −6 55 −9 −29 26−5 14 35 −23
satisfies rank(A) ≤ 3. Since rank(A) 6= # columns = 4, the list isnot linearly independent.
![Page 85: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/85.jpg)
The “Pivot Columns” of a MatrixDefinition
DefinitionThe pivot columns of a matrix A are the columns of A thatcorrespond to pivot columns in rref(A).
Example
Consider the calculation
rref
A3 −9 7−2 6 2
1 −3 −613 −39 0
=
1 −3 00 0 10 0 00 0 0
The pivot columns of A are 〈3, −2, 1, 13〉 and 〈7, 2, −6, 0〉 .
![Page 86: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/86.jpg)
The “Pivot Columns” of a MatrixDefinition
DefinitionThe pivot columns of a matrix A are the columns of A thatcorrespond to pivot columns in rref(A).
Example
Consider the calculation
rref
A3 −9 7−2 6 2
1 −3 −613 −39 0
=
1 −3 00 0 10 0 00 0 0
The pivot columns of A are 〈3, −2, 1, 13〉 and 〈7, 2, −6, 0〉 .
![Page 87: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/87.jpg)
The “Pivot Columns” of a MatrixDefinition
DefinitionThe pivot columns of a matrix A are the columns of A thatcorrespond to pivot columns in rref(A).
Example
Consider the calculation
rref
A3 −9 7−2 6 2
1 −3 −613 −39 0
=
1 −3 00 0 10 0 00 0 0
The pivot columns of A are 〈3, −2, 1, 13〉 and 〈7, 2, −6, 0〉 .
![Page 88: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/88.jpg)
The “Pivot Columns” of a MatrixDefinition
TheoremThe pivot columns of a matrix are linearly independent.
TheoremLet { #»v 1,
#»v 2, . . . ,#»v d} be the pivot columns of A. Then
Col(A) = Col([
#»v 1#»v 2 · · · #»v d
])
In particular, every column of A is a linear combination of the pivotcolumns of A.
![Page 89: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/89.jpg)
The “Pivot Columns” of a MatrixDefinition
TheoremThe pivot columns of a matrix are linearly independent.
TheoremLet { #»v 1,
#»v 2, . . . ,#»v d} be the pivot columns of A. Then
Col(A) = Col([
#»v 1#»v 2 · · · #»v d
])
In particular, every column of A is a linear combination of the pivotcolumns of A.
![Page 90: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/90.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 91: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/91.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 92: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/92.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 =
− 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 93: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/93.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1
#»a 3 = 4 · #»a 1#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 94: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/94.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 =
4 · #»a 1#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 95: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/95.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 = 7 · #»a 1 + 6 · #»a 4
![Page 96: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/96.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 =
7 · #»a 1 + 6 · #»a 4
![Page 97: Linear Independence - Math 218bfitzpat/teaching/218s20/lectures/... · Math 218 Brian D. Fitzpatrick Duke University March 3, 2020 MATH. Overview Geometric Motivation Counting \Directions"](https://reader036.fdocuments.us/reader036/viewer/2022071016/5fcf52ec0abd2137660167ba/html5/thumbnails/97.jpg)
The “Pivot Columns” of a MatrixExample
Example
Consider the calculation
rref
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
=
1 −3 4 0 70 0 0 1 60 0 0 0 0
This shows that
Col
1 −3 4 −3 −11−2 6 −8 7 28
5 −15 20 −11 −31
= Col
1 −3−2 7
5 −11
We also have
#»a 2 = − 3 · #»a 1#»a 3 = 4 · #»a 1
#»a 5 = 7 · #»a 1 + 6 · #»a 4