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

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{{12-240/Navigation}}
 
{{12-240/Navigation}}
 
'''== Linear transformation =='''
 
'''== Linear transformation =='''
'''Definition:''' A function L: V-> W is called a linear transformation if it preserve following structures:
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'''Definition:'''  
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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)
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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)
 
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)
  
'''Proof:
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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
 
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
 
2)(=>)Assume L is linear transformation
  
L(cx + y)= L(cx) + L(y)= c*L(x) + L(y)
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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
  
  (<=) 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
        2. Follows by taking y=0
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'''Examples'''
 
'''Examples'''

Revision as of 14:29, 8 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

Lecture notes uploaded by gracez