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 V}

FxV={(c,v): c F, v V}

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

Such that

VS1 x, y V: x+y = y+x

VS2 x, y, z V: x+(y+z) = (x+y)+z

VS3 x V: 0 ( of V) +x = x

VS4 x V, V V: v + x= 0 ( of V)

VS5 x V, 1 (of F) .x = x

VS6 a, b F, x V: (ab)x = a(bx)

VS7 a F, x, y V: a(x + y)= ax + ay

VS8 a, b F, x V: (a + b)x = ax + bx

Scanned Notes by Richardm