12-240/Classnotes for Tuesday September 25
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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 [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
