engg2013 unit 14 subspace and dimension mar, 2011
Post on 22-Dec-2015
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Yesterday
• Every basis in contains two vectors
• Every basis in contains three vectors
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x
y
xy
z
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 . kshum ENGG2013 3
Review of set and subset
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Cities in China
Shanghai
Beijing
Hong Kong
Tianjing
Wuhan
Guangzhou
Shenzhen
Subset of cities in Guangdongprovince
Review: Intersection and union
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F: Set of fruits
A: subset o
f fruit w
ith re
d skin
B: seedless
A union B = {cherry, apple, raspberry, watermelon}
A intersect B = {raspberry}
Subspace: definition
• A subspace W in is a subset which is– Closed under addition– Closed under scalar multiplication
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W
Intersection
• Intersection of two subpaces is also a subspace.
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x
y
z
For example, the intersectionof the x-y plane and the x-z planeis the same as the x-axis
Union • Union of two subspace is in general not a subspace.
– It is closed under scalar multiplicationbut not closed under addition.
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x
y
z
For example, the unionof the x-y plane and the z axisis not closed under addition
Lattice points
• The set is not a subspace
– It is closed under addition, – But not closed under scalar multiplication
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1
1
2
2
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.kshum ENGG2013 14
Alternate definition
• A set of k vectors
is called a basis of a subspace W in , if1.The k vectors are linearly independent2.The span of them is W.The dimension of W is defined as k.We say that W is generated by these k vectors.kshum ENGG2013 15
Example
• 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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x
yz
W
Example
• Let W be the y-axis
• The set
containing only one elementis a basis of W.
Dimension of W is 1.kshum ENGG2013 17
x
y
z
W
Question
• Let W be the y-axisshifted to the right by one unit.
• What is the dimensionof W?
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x
yz
W1