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subspaces
sittichoke somam
Mahidol Witthayanusorn School
13 สิงหาคม พ.ศ. 2555
13 สิงหาคม พ.ศ. 2555 : subspaces
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Outline
Outline
1
Subspace
Examples of subspaces
Building Subspaces
2
Linear Combination and Span
Linear Combination
Spanning
Test
13 สิงหาคม พ.ศ. 2555 : subspaces
p. 2/ 25
Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Outline
1
Subspace
Examples of subspaces
Building Subspaces
2
Linear Combination and Span
Linear Combination
Spanning
Test
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Definition
A subset W of a vector space V is called a subspace of V if W is
itself a vector space under the addition and scalar multiplication
defined on V .
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Definition
A subset W of a vector space V is called a subspace of V if W is
itself a vector space under the addition and scalar multiplication
defined on V .
Theorem
If W is a set of one or more vectors from a vector space V , then W
is a subspace of V if and only if the following conditions hold.
If u and v are vectors in W , then u + v is in W .
If k is any scalar and u is any vector in W , then ku is in W
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Example
Lines through the origin are subspaces of R2 and of R3
Planes through the origin are subspaces of R3
W = {0v }, V is any vector space.
W is the set of continuous functions, V is a function space.
the functions that are continuously differentiable on (∞, −∞)
form a subspace of F (∞, −∞)
W = {p(t) ∈ P2 such that P(2) = 0}, V = P2
W = {A ∈ Mnn |A is symmetric matric}, V = Mnn
W = {(an ) ∈ R|(an ) is convergent}, V is sequence space
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Show that W isn’t subspase of V
Example
V = R2
S = {(x, y ) ∈ R2 |x ≥ 0 and y ≥ 0}
T = {(x, y ) ∈ R2 |(x ≥ 0 and y ≥ 0) or (x ≤ 0 and y ≤ 0)}
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Theorem
If Ax = 0 is a homogeneous linear system of m equations in n
unknowns, then the set of solution vectors is a subspace of Rn .
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Example (Solution Spaces That Are Subspaces of R3 )
(1)
(2)
(3)
(4)

   
1 −2 3
x
0

   
2 −4 6 y  = 0
3 −6 9
z
0

   
1 −2 3
x
0

   
−3 7 −8 y  = 0
−2 4 −6
z
0

   
0
1 −2 3
x
   

−3 7 −8 y  = 0
4
1
2
z
0

   
0 0 0
x
0

   
0 0 0 y  = 0
0 0 0
z
0
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Subspace
Linear Combination and Span
Examples of subspaces
Building Subspaces
Theorem
If W1 , W2 , . . . , Wr are subspaces of a vector space V , then the
intersection of these subspaces is also a subspace of V .
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Outline
1
Subspace
Examples of subspaces
Building Subspaces
2
Linear Combination and Span
Linear Combination
Spanning
Test
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Definition
A vector w is called a linear combination of the vectors v1 , v2 , . . . , vn
if it can be expressed in the form
w = k1 v1 + k2 v2 + · · · + kn vn
where k1 , k2 , . . . , kn ∈ F
Example
V = R3 and i = (1, 0, 0), j = (0, 1, 0), . . . , k = (0, 0, 1)
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Theorem
If S = {w1 , w2 , . . . , wr } is a nonempty set of vectors in a vector
space V , then:
1
The set W of all possible linear combinations of the vectors in
S is a subspace of V .
2
The set W in part (a) is the “smallest” subspace of V that
contains all of the vectors in S in the sense that any other
subspace that contains those vectors contains W .
Example
Let u = (1, 2, −1), v = (6, 4, 2) be vector in R3 . Show that
w = (9, 2, 7) is a linear combination of u, v and that w 0 = (4, −1, 6) is
not a linear combination of u and v .
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Definition
If S = {v1 , v2 , . . . , vn } is a set of vectors in a vector space V , then
the subspace W of V consisting of all linear combinations of the
vectors in S is called the space spanned by v1 , v2 , . . . , vn , and we
say that the vectors S = {v1 , v2 , . . . , vn } span W . To indicate that W
is the space spanned by the vectors in the set S = {v1 , v2 , . . . , vn } ,
we write
W = span(S) or W = span{v1 , v2 , . . . , vn }
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Example
Spaces Spanned by One or Two Vectors.
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Example
Spaces Spanned by One or Two Vectors.
Example
Spanning Set for Pn .
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Example
Spaces Spanned by One or Two Vectors.
Example
Spanning Set for Pn .
Example
Determine whether v1 = (1, 1, 2), v2 = (1, 0, 1), and v3 = (2, 1, 3) span
the vector space R3 .
13 สิงหาคม พ.ศ. 2555 : subspaces
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Theorem
If S = {v1 , v2 , . . . , vn } and S 0 = {w1 , w2 , . . . , wk } are two sets of
vectors in a vector space V , then
span{v1 , v2 , . . . , vn } = span{w1 , w2 , . . . , wk }
if and only if each vector in S is a linear combination of those in S 0
and each vector in S 0 is a linear combination of those in S.
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Theorem
If V is a vector space over F and v1 , v2 , . . . , vk , vk+1 are vector in V ,
then span{v1 , v2 , . . . , vk } ⊂ span{v1 , v2 , . . . , vk+1 }
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Theorem
Let v1 , v2 , . . . , vk be a vector space vector F
span{v1 , v2 , . . . , vk } = span{v1 , v2 , . . . , vk+1 } ↔ vk+1 ∈ span{v1 , v2 , . . . , vk }
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Which of the following are linear combinations of u = (0, −2, 2) and
v = (1, 3, −1)
1
(2, 2, 2)
2
(3, 1, 5)
3
(0, 4, 5)
4
(0, 0, 0)
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Which of the following are linear combinations of
"
#
"
#
"
4
0
1 −1
0
A=
,B =
,C =
−2 −2
2 3
1
#
2
4
"
1
2
3
4
#
6 −8
−1 −8
"
#
0 0
0 0
"
#
6 0
3 8
"
#
−1 5
7 1
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
In each part express the vector as a linear combination of
P1 = 2 + x + 4x 2 , P2 = 1 − x + 3x 2 and P3 = 3 + 2x + 5x 2
1
−9 + 2x + 5x 2
2
6 + 11x + 6x 2
3
0
4
7 + 8x + 9x 2
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
In each part, determine whether the given vectors span R3
1
v1 = (2, 2, 2), v2 = (0, 0, 3), v3 = (0, 1, 1)
2
v1 = (2, −1, 3), v2 = (4, 1, 2), v3 = (8, −1, 8)
3
v1 = (3, 1, 4), v2 = (2, −3, 5), v3 = (5, −2, 9), v4 = (1, 4, −1)
4
v1 = (1, 2, 6), v2 = (3, 4, 1), v3 = (4, 3, 1), v4 = (3, 3, 1)
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Suppose that v1 = (2, 1, 0, 3), v2 = (3, −1, 5, 2) and v3 = (−1, 0, 2, 1)
Which of the following vectors are in Span{v1 , v2 , v3 }?
1
(2, 3, −7, 3)
2
(0, 0, 0, 0)
3
(1, 1, 1, 1)
4
(−4, 6, −13, 4)
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Determine whether the following polynomials SpanP2
p1 = 1 − x + 2x 2 , p2 = 3 + x, p3 = 5 − x + 4x 2 , p4 = −2 − 2x + 2x 2
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Subspace
Linear Combination and Span
Linear Combination
Spanning
Test
Let f = cos2 x and g = sin2 x. Which of the following lie in the space
spanned of f and g ?
1
cos 2x
2
3 + x2
3
1
4
sin x
5
0
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