Linear IndependenceMath 218
Brian D. Fitzpatrick
Duke University
March 3, 2020
MATH
Overview
Geometric MotivationCounting “Directions”
BackgroundLinear Combinations as Matrix MultiplicationAn Important Observation
Definitions and ExamplesDefinitionsExamples
The Linear Independence TestStatementExample
The “Pivot Columns” of a MatrixDefinitionExample
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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〉 }.
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.
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.
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.
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.
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
]
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
]
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?
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?
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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〉 .
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〉 .
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〉 .
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.
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.
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
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
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
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
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
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
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
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