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: | + | |
+ | '''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 15:29, 8 November 2012
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== 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)