12-240/Classnotes for Tuesday September 25

DBN: Classes: 2012-13: 12-240 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 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) Register of Good Deeds Add your name / see who's in!

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 [math]\displaystyle{ \in\!\, }[/math] V}

FxV={(c,v): c [math]\displaystyle{ \in\!\, }[/math] F, v [math]\displaystyle{ \in\!\, }[/math] V}

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

Such that

VS1 [math]\displaystyle{ \forall\!\, }[/math] x, y [math]\displaystyle{ \in\!\, }[/math] V: x+y = y+x

VS2 [math]\displaystyle{ \forall\!\, }[/math] x, y, z [math]\displaystyle{ \in\!\, }[/math] V: x+(y+z) = (x+y)+z

VS3 [math]\displaystyle{ \forall\!\, }[/math] x [math]\displaystyle{ \in\!\, }[/math] V: 0 ( of V) +x = x

VS4 [math]\displaystyle{ \forall\!\, }[/math] x [math]\displaystyle{ \in\!\, }[/math] V, [math]\displaystyle{ \exists \!\, }[/math] V [math]\displaystyle{ \in\!\, }[/math] V: v + x= 0 ( of V)

VS5 [math]\displaystyle{ \forall\!\, }[/math] x [math]\displaystyle{ \in\!\, }[/math] V, 1 (of F) .x = x

VS6 [math]\displaystyle{ \forall\!\, }[/math] a, b [math]\displaystyle{ \in\!\, }[/math] F, [math]\displaystyle{ \forall\!\, }[/math] x [math]\displaystyle{ \in\!\, }[/math] V: (ab)x = a(bx)

VS7 [math]\displaystyle{ \forall\!\, }[/math] a [math]\displaystyle{ \in\!\, }[/math] F, [math]\displaystyle{ \forall\!\, }[/math] x, y [math]\displaystyle{ \in\!\, }[/math] V: a(x + y)= ax + ay

VS8 [math]\displaystyle{ \forall\!\, }[/math] a, b [math]\displaystyle{ \in\!\, }[/math] F, [math]\displaystyle{ \forall\!\, }[/math] x [math]\displaystyle{ \in\!\, }[/math] V: (a + b)x = ax + bx

Scanned Notes by Richardm

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