12-240/Classnotes for Tuesday September 25

#	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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Today's class dealt with the properties of vector spaces.

Definition

Let F is a field, a vector space V over F is a set V of vectors with special element O ( of V) and tow operations: (+): VxV->V, (.): FxV->V

VxV={(v,w): v,w $\in \!\,$ V}

FxV={(c,v): c $\in \!\,$ F, v $\in \!\,$ V}

Then, (+): VxV->V is (v,w)= v+w; (.): FxV->V is (c,v)=cv

Such that

VS1 $\forall \!\,$ x, y $\in \!\,$ V: x+y = y+x

VS2 $\forall \!\,$ x, y, z $\in \!\,$ V: x+(y+z) = (x+y)+z

VS3 $\forall \!\,$ x $\in \!\,$ V: 0 ( of V) +x = x

VS4 $\forall \!\,$ x $\in \!\,$ V, $\exists \!\,$ V $\in \!\,$ V: v + x= 0 ( of V)

VS5 $\forall \!\,$ x $\in \!\,$ V, 1 (of F) .x = x

VS6 $\forall \!\,$ a, b $\in \!\,$ F, $\forall \!\,$ x $\in \!\,$ V: (ab)x = a(bx)

VS7 $\forall \!\,$ a $\in \!\,$ F, $\forall \!\,$ x, y $\in \!\,$ V: a(x + y)= ax + ay

VS8 $\forall \!\,$ a, b $\in \!\,$ F, $\forall \!\,$ x $\in \!\,$ V: (a + b)x = ax + bx