12-240/Classnotes for Thursday September 27: Difference between revisions

Revision as of 21:26, 27 September 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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Vector Spaces

(Quick recap)

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse -> -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

1.a x + z = y + z => x = y

1.b ax = ay, a != 0, => x = y

1.c ax = bx, x != 0, => a = b

2. 0 is unique.

3. Additive inverse is unique.

4. 0_F * x = 0_V

5. a * 0_V = 0_V

6. (-a) x = -(ax) = a(-x)

7. cx = 0 <=> c = 0 or x = 0_V

@@ Line 1: / Line 1: @@
 {{12-240/Navigation}}
+'''Vector Spaces'''
-In this course, we will be focusing on both a practical side and a theoretical side.
 == Vector space axioms ==
 ''(Quick recap)''
 VS1.  x + y = y + x
 VS2.  (x + y) + z = x + (y + z)
 VS3.  0 vector
 VS4. + inverse -> -
 VS5. 1x = x
 VS6. a(bx) = (ab)x
 VS7. a(x + y) = ax + ay
 VS8. (a+b)x = ax + bx
+== Theorems ==
+.a x + z = y + z => x = y
+.b ax = ay, a != 0, => x = y
+.c ax = bx, x != 0, => a = b
+. 0 is unique.
+. Additive inverse is unique.
+. 0_F * x = 0_V
+. a * 0_V = 0_V
+. (-a) x = -(ax) = a(-x)
+. cx = 0 <=> c = 0 or x = 0_V