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

Revision as of 14:29, 8 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 =='''
-'''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)
@@ Line 15: / Line 18: @@
 ) 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:'''
-'''
 )  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)
+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
-   (<=) 1. Follows from L(c*x+y) = c*L(x)+L(y) by taking c=1
+. Follows by taking y=0
-. Follows by taking y=0
 '''Examples'''