Difference between revisions of "12240/Classnotes for Thursday October 18"
From Drorbn
(→Linear transformation) 
(→Linear transformation) 

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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)  
+  
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:  +  '''Proposition:''' 
+  
+  1) property 2 => property 3  
+  
+  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:  
+  '''  
+  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 [[User:starashstarash]]==  == lecture note on oct 18, uploaded by [[User:starashstarash]]== 
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 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)