1.2. Vecto r spaces 13 equivalent t o p(x) ®q(y) i s mappe d t o p(x)q(y). I t follow s tha t 4 induces a map of the quotient spac e Y ® ^ ont o F[jc,y ] (1.2.12) 5 : £/,,.(* ) ® ^.(y) ~ £,-(*)?, • (?)• 7 7 I i s a n isomorphism , an d th e space s Y ® ^ an d F[x,y ] ar e isomor - phic. EXERCISES FOR SECTION 1. 2 exl.2.1 Verif y tha t R is a vector space over Q, and that C is a vector space over either Q or R. exl.2.2 Le t Y:J — 1,2,3, b e vecto r space s ove r th e sam e field F . Le t (px : Y x i— • Y 2 an d (p 2 : ^ H ^ ^ b e isomorphisms . Prov e that (p 2 P! is a n isomorphism o f Y 1 ont o Y 3 . Conclud e tha t isomorphism is an equivalenc e relation for vector spaces (defined over the same field F). exl.2.3 Verif y that the intersection of subspaces is a subspace. exl.2.4 Le t $/ and W b e subspaces of a vector space Y, an d neither of them contains the other. Show that W U W i s not a subspace. Hint: Tak e u e % \ W, w e IV \ °tt and consider u + w. exl.2.5 Verif y that the sum of subspaces is a subspace, and prove that £ ^ . = span[U^] . exl.2.6 Chec k tha t fo r ever y E C Y, span[£ ] i s a subspace o f ^ , an d is contained in every subspace that contains E. exl.2.7 I f Yx is a subspace of Y an d (p is an isomorphism of Y ont o ^, the n (pYx is a subspace of W. exl.2.8 Describ e all the complements in R2 of the subspace X = {(i,0):x G R}. exl.2.9 Prov e that two subspaces Yx and Y2 in a vector space are independent if ^^ = {0}. exl.2.10 Prov e that the subspaces W}• c Y, j = 1,.. . ,N ar e independent if and only if Wj n £¥ • ^ = {0 } for all j.

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