Transcript Subspace and dimension

ENGG2013 Unit 14
Subspace and dimension
Mar, 2011.
Yesterday
• Every basis in
contains two vectors
y
x
• Every basis in
contains three vectors
z
y
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Basis: Definition
• For any given vector
in
if there is one and only one choice for the
coefficients c1, c2, …,ck, such that
we say that these k vectors form a basis of
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Review of set and subset
Cities in China
Tianjing
Beijing
Wuhan
Shanghai
Guangzhou
Shenzhen
Hong Kong
Subset of cities in Guangdong
province
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Review: Intersection and union
A union B =
{cherry, apple, raspberry, watermelon}
F: Set of fruits
A intersect B = {raspberry}
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Subspace: definition
• A subspace W in
is a subset which is
– Closed under addition
– Closed under scalar multiplication
W
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Conceptual illustration
W
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Example of subspace
• The z-axis
z
y
x
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Example of subspace
• The x-y plane
z
y
x
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Non-example
• Parabola
y
x
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Intersection
• Intersection of two subpaces is also a
subspace.
z
y
x
For example, the intersection
of the x-y plane and the x-z plane
is the same as the x-axis
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Union
• Union of two subspace is in general not a subspace.
– It is closed under scalar multiplication
but not closed under addition.
z
y
x
For example, the union
of the x-y plane and the z axis
is not closed under addition
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Lattice points
• The set
a subspace
is not
– It is closed under addition,
– But not closed under scalar multiplication
2
1
1
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Subspace, Basis and dimension
• Let W be a subspace in
• For any given vector
in W,
if there is one and only one choice for the coefficients c1,
c2, …,ck, such that
we say that these k vectors form a basis of W.
and define the dimension of subspace W by dim(W)=k.
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Alternate definition
• A set of k vectors
is called a basis of a subspace W in
, if
1.The k vectors are linearly independent
2.The span of them is W.
The dimension of W is defined as k.
We say that W is generated by these k vectors.
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Example
z
• Let W be the x-z plane
• W is a subspace
• u and v form a basis
of W.
• The dimension of W is 2.
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y
W
x
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Example
• Let W be the y-axis
z
y
W
• The set
x
containing only one element
is a basis of W.
Dimension of W is 1.
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Question
• Let W be the y-axis
shifted to the right by one unit.
z
y
1
W
• What is the dimension
of W?
x
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Question
• Let W be the straight line x=y=z.
• What is the dimension of W?
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Question
• Find a basis for the plane
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Question
• Find a basis for the intersection of
(This is the intersection of two planes:
x – 2y – z = 0, and x + y + z = 0.)
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