12240/Classnotes for Thursday October 18
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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)