10 1. Vecto r Spaces b. symmetric: x = y = y = x, sinc e x — y G W i f an d onl y i f y — x — — (x — y) e W, c. transitive: If both x~y an d y = z, then jt = z. This follows from: if both (x—y) and (y — z) are in ^, thensois x — z = ( x —y) + (y — z). This equivalenc e relatio n partition s y int o equivalenc e classes , called th e cosets o f y ^ i n y. Fo r x G f, th e coset o f ^ that con - tains x is the set x = x + W — {v = x + w : we W} —the "translate " of W b y x. We define the quotient space y' jW t o be the space whose elements are the cosets of W i n y, an d the operations of addition and multipli- cation by scalars are defined a s follows. If x — x-\-W an d y — y + W are cosets, and a G F, then (1.2.6) x + y = x + y + W=x + y an d ax —ax. The definitio n need s justification . W e define d th e su m o f tw o cosets b y takin g a representative elemen t fro m each , an d takin g th e coset that contains their sum as the sum of the cosets. We need to show that the result is well defined, i.e., that it does not depend on the choice of the representatives in the cosets. I n other words, we need to verif y that if x = x x (mo d W) an d y = y x (mo d W), the n x + y = xx + y x (mod W). Now , i f x = xx + w, y = y x + w' wit h w , w' G W, the n x + y — xx + w + yj + w f = x x + y{ + w + w\ an d sinc e w + wf G #", we havex + y = Xj+y 1 (mo d ^) . The definition of ax is justified similarly: if x = xl (mo d ^ ) , then ax — axx — a(x — xx), an d sinc e W i s a subspace, i t is closed unde r multiplication by scalars, a(x — x2) G ^, and ax = ox ^ (mo d #^ ) . 1.2.5 Direc t sums. If y[,..., y k ar e vector spaces over F, tf?e (formal) direct sum k ®r j = y 1 ®...@v k i

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