12-240/Classnotes for Thursday October 18: Difference between revisions
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1) property 2 leads to property 3 |
1) property 2 leads to property 3 |
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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) |
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) |
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== lecture note on oct 18, uploaded by [[User:starash|starash]]== |
== lecture note on oct 18, uploaded by [[User:starash|starash]]== |
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Revision as of 23:49, 29 October 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 leads to property 3 2) L: V -> W is a linear transformation iff c , 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) L(cx + y)= L(cx) + L(y)= c.L(x) + L(y)
lecture note on oct 18, uploaded by starash
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