MATH 116 - WEEK 8
Sibel Özkan
1
Gebze Technical University
April 12, 2022
1
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Today
Span of a Set of Vectors
(MATH 116 - LINEAR ALGEBRA)
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Today
Span of a Set of Vectors
Linear Independence
(MATH 116 - LINEAR ALGEBRA)
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Today
Span of a Set of Vectors
Linear Independence
Basis and Dimension
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
Definition:
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Given the set of vectors
 
 
 
−1
−1
1
u =  11  , v =  3  , w =  1 
2
2
−2
Is the vector u is in the Span {v, w} ?
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Given the set of vectors
 
 
 
−1
−1
1
u =  11  , v =  3  , w =  1 
2
2
−2
Is the vector u is in the Span {v, w} ?
Solution: Since
 
 
 
−1
−1
1
 11  = 3  3  + 2  1 
2
2
−2
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Given the set of vectors
 
 
 
−1
−1
1
u =  11  , v =  3  , w =  1 
2
2
−2
Is the vector u is in the Span {v, w} ?
Solution: Since
 
 
 
−1
−1
1
 11  = 3  3  + 2  1 
2
2
−2
Yes.
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: In R3 , the span of e1 and e2 is the set of vectors of the form
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: In R3 , the span of e1 and e2 is the set of vectors of the form
 
α
αe1 + βe2 =  β
0
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
As an exercise, verify that Span(e1 , e2 ) is a subspace of R3
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
As an exercise, verify that Span(e1 , e2 ) is a subspace of R3
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
As an exercise, verify that Span(e1 , e2 ) is a subspace of R3
Also the span of e1 , e2 , e3 is the set of all vectors of the form
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
As an exercise, verify that Span(e1 , e2 ) is a subspace of R3
Also the span of e1 , e2 , e3 is the set of all vectors of the form
 
α1
α 1 e1 + α 2 e2 + α 3 e3 =  α 2 
α3
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
As an exercise, verify that Span(e1 , e2 ) is a subspace of R3
Also the span of e1 , e2 , e3 is the set of all vectors of the form
 
α1
α 1 e1 + α 2 e2 + α 3 e3 =  α 2 
α3
So it is all of R3 .
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
Theorem
If v1 , v2 , . . . , vn are elements of a vector space V , then
Span(v1 , v2 , . . . , vn ) is a subspace of V .
(MATH 116 - LINEAR ALGEBRA)
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Span of a Set of Vectors
Theorem
If v1 , v2 , . . . , vn are elements of a vector space V , then
Span(v1 , v2 , . . . , vn ) is a subspace of V .
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Think of the following vectors in R3
 
 
 
1
−2
−1
x1 = −1 , x2 =  3  , x3 =  3 
2
1
8
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Think of the following vectors in R3
 
 
 
1
−2
−1
x1 = −1 , x2 =  3  , x3 =  3 
2
1
8
Let S be the subspace of R3 spanned by x1 , x2 , x3 .
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Linear Independence
Think of the following vectors in R3
 
 
 
1
−2
−1
x1 = −1 , x2 =  3  , x3 =  3 
2
1
8
Let S be the subspace of R3 spanned by x1 , x2 , x3 .
S can be represented in terms of only x1 and x2 since x3 is already in the span of
x1 and x2 :
x3 = 3x1 + 2x2
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Think of the following vectors in R3
 
 
 
1
−2
−1
x1 = −1 , x2 =  3  , x3 =  3 
2
1
8
Let S be the subspace of R3 spanned by x1 , x2 , x3 .
S can be represented in terms of only x1 and x2 since x3 is already in the span of
x1 and x2 :
x3 = 3x1 + 2x2
So, any linear combination of x1 , x2 and x3 cab be reduced to the linear
combination of x1 and x2 :
α1 x1 + α2 x2 + α3 x3 = α1 x1 + α2 x2 + α3 (3x1 + 2x2 ) = (α1 + 3α3 )x1 + (α2 + α3 )x2
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Think of the following vectors in R3
 
 
 
1
−2
−1
x1 = −1 , x2 =  3  , x3 =  3 
2
1
8
Let S be the subspace of R3 spanned by x1 , x2 , x3 .
S can be represented in terms of only x1 and x2 since x3 is already in the span of
x1 and x2 :
x3 = 3x1 + 2x2
So, any linear combination of x1 , x2 and x3 cab be reduced to the linear
combination of x1 and x2 :
α1 x1 + α2 x2 + α3 x3 = α1 x1 + α2 x2 + α3 (3x1 + 2x2 ) = (α1 + 3α3 )x1 + (α2 + α3 )x2
So, S= Span(x1 , x2 , x3 ) = Span(x1 , x2 )
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0
Since the coefficients are not zero, we can solve for any vector in terms of the
other two:
x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0
Since the coefficients are not zero, we can solve for any vector in terms of the
other two:
x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2
So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 )
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0
Since the coefficients are not zero, we can solve for any vector in terms of the
other two:
x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2
So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 )
On the otherhand, there is no such dependency relation between x1 and x2 .
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
The above equation can be rewritten as 3x1 + 2x2 − 1x3 = 0
Since the coefficients are not zero, we can solve for any vector in terms of the
other two:
x1 = (−2/3)x2 + (1/3)x3 , x2 = (−3/2)x1 + (1/2)x3 , x3 = 3x1 + 2x2
So, Span(x1 , x2 , x3 ) = Span(x2 , x3 ) = Span(x1 , x3 ) = Span(x1 , x2 )
On the otherhand, there is no such dependency relation between x1 and x2 .
So, if c1 x1 + c2 x2 = 0, only solution is c1 = c2 = 0
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
We can generalize these observations as:
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Definition:
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Definition:
1
1
Example: The vectors
and
are linearly independent.
1
2
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Definition:
1
1
Example: The vectors
and
are linearly independent.
1
2
Since if
1
1
0
c1
+ c2
=
1
2
0
(MATH 116 - LINEAR ALGEBRA)
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Linear Independence
Definition:
1
1
Example: The vectors
and
are linearly independent.
1
2
Since if
1
1
0
c1
+ c2
=
1
2
0
then
c1 + c2 = 0,
c1 + 2c2 = 0
and only solution to the system is c1 = c2 = 0
(MATH 116 - LINEAR ALGEBRA)
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Linear Dependence
Definition:
(MATH 116 - LINEAR ALGEBRA)
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Linear Dependence
Definition:
Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since
(MATH 116 - LINEAR ALGEBRA)
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Linear Dependence
Definition:
Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since
e1 + 2e2 + 3e3 − x = 0
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Linear Dependence
Definition:
Example: Let x = (1, 2, 3)T . The vectors e1 , e2 , e3 and x are linearly dependent since
e1 + 2e2 + 3e3 − x = 0
(See in this case c1 = 1, c2 = 2, c3 = 3, c4 = −1)
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Geometric Interpretation
(MATH 116 - LINEAR ALGEBRA)
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Geometric Interpretation
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Example
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
Solution: We must show that all c are zeros below.
c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
Solution: We must show that all c are zeros below.
c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T
We have
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
Solution: We must show that all c are zeros below.
c1 (1, 1, 1)T + c2 (1, 1, 0)T + c3 (1, 0, 0)T = (0, 0, 0)T
We have
which gives us c1 = 0, c2 = 0, c3 = 0. Answer is YES.
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent?
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent?
Solution: We must show that all c’s are zeros
c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent?
Solution: We must show that all c’s are zeros
c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T
which is
(c1 , c2 , c1 )T = (0, 0, 0)T
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (1, 0, 1)T , (0, 1, 0)T linearly independent?
Solution: We must show that all c’s are zeros
c1 (1, 0, 1)T + c2 (0, 1, 0)T = (0, 0, 0)T
which is
(c1 , c2 , c1 )T = (0, 0, 0)T
The answer is YES.
(MATH 116 - LINEAR ALGEBRA)
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Theorem
Theorem
Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The
vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular.
(MATH 116 - LINEAR ALGEBRA)
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Theorem
Theorem
Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The
vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular.
Proof: The equation
c1 x1 + c2 x2 + · · · + cn xn = 0
can be written as the matrix equation
(MATH 116 - LINEAR ALGEBRA)
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Theorem
Theorem
Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The
vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular.
Proof: The equation
c1 x1 + c2 x2 + · · · + cn xn = 0
can be written as the matrix equation
Xc = 0
(MATH 116 - LINEAR ALGEBRA)
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Theorem
Theorem
Let x1 , x2 , . . . , xn be n vectors in Rn and let X = (x1 x2 . . . xn ). The
vectors x1 , x2 , . . . , xn are linearly dependent if and only if X is singular.
Proof: The equation
c1 x1 + c2 x2 + · · · + cn xn = 0
can be written as the matrix equation
Xc = 0
This equation has non-trivial (non-zero) solutions if and only if X is
singular.
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent?
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent?
Solution: Since
4 2 2
2 3 −5 = 0
3 1 3
(MATH 116 - LINEAR ALGEBRA)
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Example
Example: Is the set (4, 2, 3)T , (2, 3, 1)T , (2, −5, 3)T linearly independent?
Solution: Since
4 2 2
2 3 −5 = 0
3 1 3
NO: The vectors are linearly dependent.
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
Solution: Since
1 1 1
0 1 1 =1
0 0 1
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T linearly independent?
Solution: Since
1 1 1
0 1 1 =1
0 0 1
YES. The vectors are linearly independent.
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly
independent?
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly
independent?
Solution: Let’s put them in Augmented Matrix form
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly
independent?
Solution: Let’s put them in Augmented Matrix form
(MATH 116 - LINEAR ALGEBRA)
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Example
:
Example: Is the set (1, −1, 2, 3)T , (−2, 3, 1, −2)T , (1, 0, 7, 7)T linearly
independent?
Solution: Let’s put them in Augmented Matrix form
Since the REF involves a free variable c3 , there are nontrivial solutions,
hence the vectors are linearly dependent.
(MATH 116 - LINEAR ALGEBRA)
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Theorem
Theorem
Let v1 , v2 , . . . , vn be vectors in a vector space V . A vector
v ∈ Span (v1 , v2 , . . . , vn ) can be written uniquely as a linear combination
of v1 , v2 , . . . , vn if and only if v1 , v2 , . . . , vn are linearly independent.
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Definition
The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if
(i) v1 , v2 , . . . , vn are linearly independent.
(ii) v1 , v2 , . . . , vn span V .
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Definition
The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if
(i) v1 , v2 , . . . , vn are linearly independent.
(ii) v1 , v2 , . . . , vn span V .
Example: The standard basis for R3 is {e1 , e2 , e3 }; but there are many bases we could
choose for R3 . Here are some examples:
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Definition
The vectors v1 , v2 , . . . , vn form a basis for a vector space V if and only if
(i) v1 , v2 , . . . , vn are linearly independent.
(ii) v1 , v2 , . . . , vn span V .
Example: The standard basis for R3 is {e1 , e2 , e3 }; but there are many bases we could
choose for R3 . Here are some examples:
(MATH 116 - LINEAR ALGEBRA)
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Example
(MATH 116 - LINEAR ALGEBRA)
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A Theorem and a Corollary
Theorem
If v1 , v2 , . . . , vn is a spanning set for a vector space V , then any collection
of m vectors in V , where m > n, is linearly dependent.
(MATH 116 - LINEAR ALGEBRA)
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A Theorem and a Corollary
Theorem
If v1 , v2 , . . . , vn is a spanning set for a vector space V , then any collection
of m vectors in V , where m > n, is linearly dependent.
Corollary
If both v1 , v2 , . . . , vn and u1 , u2 , . . . , um are basis for a vector space V ,
then n = m.
(MATH 116 - LINEAR ALGEBRA)
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Dimension
(MATH 116 - LINEAR ALGEBRA)
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Dimension
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) any set of n linearly independent vectors spans V
(ii) any n vectors that span V are linearly independent.
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) any set of n linearly independent vectors spans V
(ii) any n vectors that span V are linearly independent.
Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 .
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) any set of n linearly independent vectors spans V
(ii) any n vectors that span V are linearly independent.
Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 .
Solution: Since dimR3 = 3, we only need to show that they are linearly independent.
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) any set of n linearly independent vectors spans V
(ii) any n vectors that span V are linearly independent.
Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 .
Solution: Since dimR3 = 3, we only need to show that they are linearly independent.
Since
1 −2 1
2
1
0 =2
3
0
1
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) any set of n linearly independent vectors spans V
(ii) any n vectors that span V are linearly independent.
Example: Show that (1, 2, 3)T , (−2, 1, 0)T , (1, 0, 1)T is a basis for R3 .
Solution: Since dimR3 = 3, we only need to show that they are linearly independent.
Since
1 −2 1
2
1
0 =2
3
0
1
they are linearly independent and this set is a basis for R3 .
(MATH 116 - LINEAR ALGEBRA)
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Basis and Dimension
Theorem
If V is a vector space of dimension n > 0, then
(i) no set of fewer than n vectors can span V
(ii) any subset of fewer than n linearly independent vectors can be extended to form a
basis for V
(iii) any spanning set containing more than n vectors can be pared down to form a
basis for V .
(MATH 116 - LINEAR ALGEBRA)
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David C. Lay, Linear Algebra and its Applications, Addison-Wesley
(2012)
S. J. Leon, Linear Algebra with Applications, Pearson (2014)
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