12-240/Classnotes for Tuesday September 25

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


Contents

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


Examples

Properties

Scanned Notes by Richardm