12-240/Classnotes for Thursday October 18: Difference between revisions

Revision as of 14:31, 2 November 2012

#	Week of...	Notes and Links
Additions to this web site no longer count towards good deed points.
1	Sep 10	About This Class, Tuesday, Thursday
2	Sep 17	HW1, Tuesday, Thursday, HW1 Solutions
3	Sep 24	HW2, Tuesday, Class Photo, Thursday
4	Oct 1	HW3, Tuesday, Thursday
5	Oct 8	HW4, Tuesday, Thursday
6	Oct 15	Tuesday, Thursday
7	Oct 22	HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
8	Oct 29	Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
9	Nov 5	Tuesday, Thursday
10	Nov 12	Monday-Tuesday is UofT November break, HW7, Thursday
11	Nov 19	HW8, Tuesday,Thursday
12	Nov 26	HW9, Tuesday , Thursday
13	Dec 3	Tuesday UofT Fall Semester ends Wednesday
F	Dec 10	The Final Exam (time, place, style, office hours times)
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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 $\forall \,\!$ c $\in \,\!$ F, $\forall \,\!$ x, y $\in \,\!$ 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)

@@ Line 1: / Line 1: @@
 {{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:
 ) L(x + y)= L(x) + L(y)
 ) L(cx)= c.L(x)
 ) L(0 of V) = 0 of W
-Proposition:
+'''Proposition:'''
+) property 2 => property 3
+) 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:
+'''
+)  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
+)(=>)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
+. Follows by taking y=0
+'''Examples'''
+. L: '''R'''^2 -> '''R'''^2 by
+. P,Q: P(F)
-) property 2 leads to property 3
-) 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)
-Proof:
-)  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
-) L(cx + y)= L(cx) + L(y)= c.L(x) + L(y)
 == lecture note on oct 18, uploaded by [[User:starash|starash]]==