Difference between revisions of "12240/Classnotes for Thursday October 18"
From Drorbn
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{{12240/Navigation}}  {{12240/Navigation}}  
'''== Linear transformation =='''  '''== Linear transformation =='''  
−  '''Definition:''' A function L: V> W is called a linear transformation if it preserve following structures:  +  
+  '''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)  
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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:  +  '''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  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)  
+  
+  (<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1  
−  +  2. Follows by taking y=0  
−  +  
'''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 c F, x, y 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)