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Chapter 1

Prop 1.9

V = U ⊕W if and only if V = U +W and U ∩W = {0}

Prop 1.8

(a) V = U1 + · · · + Un; (b) the only way to write 0 as a sum u1 + · · · + un, where each uj ∈ Uj , is by taking all the uj’s equal to 0.

Prop 1.7

U +W = {(x,y, 0) : x,y ∈ F}

Prop 1.6

v + (−1)v = 1v + (−1)v = 1 + (−1)v = 0v = 0

Prop 1.5

a0 = a(0 + 0) = a0 + a0

Prop 1.4

0v = (0 + 0)v = 0v + 0v

Prop 1.3

w=w+0=w+(v+w′)=(w+v)+w′ =0+w′ =w′

Prop 1.2

0′ = 0′ + 0 = 0