Difference between revisions of "12-240/Classnotes for Thursday October 18"

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(Linear transformation)
(Linear transformation)
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{{12-240/Navigation}}
 
{{12-240/Navigation}}
== Linear transformation ==
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'''== Linear transformation =='''
Definition: A function L: V-> W is called a linear transformation if it preserve following structures:
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'''Definition:''' A function L: V-> W is called a linear transformation if it preserve following structures:
  
 
1) L(x + y)= L(x) + L(y)
 
1) L(x + y)= L(x) + L(y)
 +
 
2) L(cx)= c.L(x)
 
2) L(cx)= c.L(x)
 +
 
3) L(0 of V) = 0 of W
 
3) L(0 of V) = 0 of W
  
Proposition:
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'''Proposition:'''
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1) property 2 => property 3
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2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math> F, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)
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 +
'''Proof:
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'''
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1)  take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F * 0 of V)=(0 of F)*L(0 of V)=0 of W
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 +
2)(=>)Assume L is linear transformation
 +
 
 +
L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)
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 +
  (<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1
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        2. Follows by taking y=0
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'''Examples'''
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1. L: '''R'''^2 -> '''R'''^2 by
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1) property 2 leads to property 3
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2. P,Q: P(F)
2) L: V -> W is a linear transformation iff <math>\forall\,\!</math> c <math>\in\,\!</math>, <math>\forall\,\!</math> x, y <math>\in\,\!</math> V: L(cx + y)= cL(x) + L(y)
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Proof:
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1)  take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F . 0 of V)=(0 of F).L(0 of V)=0 of W
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2) L(cx + y)= L(cx) + L(y)= c.L(x) + L(y)
+
  
 
== lecture note on oct 18, uploaded by [[User:starash|starash]]==
 
== lecture note on oct 18, uploaded by [[User:starash|starash]]==

Revision as of 13:31, 2 November 2012

== Linear transformation == Definition: A function L: V-> W is called a linear transformation if it preserve following structures:

1) L(x + y)= L(x) + L(y)

2) L(cx)= c.L(x)

3) L(0 of V) = 0 of W

Proposition:

1) property 2 => property 3

2) L: V -> W is a linear transformation iff \forall\,\! c \in\,\! F, \forall\,\! x, y \in\,\! V: L(cx + y)= cL(x) + L(y)

Proof: 1) take c= 0 in F and x=0 in V. Then L(cx)=cL(x) -> L(0 of F * 0 of V)=(0 of F)*L(0 of V)=0 of W

2)(=>)Assume L is linear transformation

L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)
  (<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1
       2. Follows by taking y=0

Examples

1. L: R^2 -> R^2 by


2. P,Q: P(F)

lecture note on oct 18, uploaded by starash