12-240/Classnotes for Tuesday October 2: Difference between revisions

Revision as of 11:58, 4 October 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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The "vitamins" slide we viewed today is here.

Today, the professor introduces more about subspace, linear combination, and related subjects.

Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and contain 0 of V

Proof:

First direction:

if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .

=> + W is closed under the operations of V.

+ W has a unique identity of addition:  $\forall \!\,$  a  $\in \!\,$  W: 0 + a = a

Moreover, a a $\in \!\,$ V. Hence 0 is also identity of addtition of V

Second direction

if a non-empty subset W ⊂ V is closed under the operations of V

we need to prove that

@@ Line 14: / Line 14: @@
 First direction:
- if  a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .
+if  a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .
 => + W is closed under the operations of V.
-   + W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a
+ + W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a
+Moreover, a a <math>\in\!\,</math> V. Hence 0 is also identity of addtition of V
 Second direction