664
VECTOR SPACES
a1 1 k'1 +... + alkx:k = b1..., (amlx-I +. ' + (1rn.iti ba
isther111(empl1/tyt or ( islnear1 varie(tyt o ffV..
Remark. We must allow\ for tlhe possibility that X is emprlty sintce we could
have all the a(I1=O(and some b.f:0'then no 10l,... x ) would satisfy the
The proof is left as anl exercise (P robilem 5).,
PROBLEMS
1. Represenrt each of the followxing as a linear v ar aiyspefcifying ' a adr, nd
dlescribing the base.,space. 'Thlrou h out. -so <1;s, - o s so
(a) n1V2: all,(x. y) such that.x 1 ~ 31, ify-2 -S5.
() In Vt::a=ll (x, y) suich that x.= 3 - 5t q 7 - 21.
(c) hi V2: all (, y) sutch that 3')t - 2q= 6
(d) In V2: all (xy) surch that 2x +~y= j
(e) InXV: all(;xy,)su cthat x... - t1j = 1+ t...3~1.
l(? In-1\V3: a l l (x y such thatx. = 7+1' =2}- t+1 I,
()InI V-: all (xvyI/, uch that xs + y + 3.
(h)In V:1'rall '(x'.Y,;?such that 2.v + 3y+41 12.
(i) InIV 3:,all.v y1,)suchthatx=2+,~ t~2s - t-.4 4 +1t-s,.
(j) }InV 3:all.4xz) s uh ta t= -1--7.1 -2 --5.s, t + S.
(k) In V 4: all (x, x2,x,x4) such that x1 1 +~t, x 22-t, = _f-t
x 2-+?t.
(1) In V r all ( 1.x3x} surch that X t._3 -1 '2= s. 4~1 x, =5 +1t
=2+ 1.
(m)InV2: all (, y) schthatx=s 1+1~t y2-f+-s
3tn)InV3all jy,: scthatx 1y = 2 =3
(o) In Vfall (x, ),,such that x=2+2s y,1 +41te5
()lin V4: all (~~~4 such that x = 2+4 t,-v === 3+ t-s,
lit =,5a-li x4,5.a,t, tc- th tx I+.s,,x4..
(r) In \Vt: all (.vx2, x3, x) suehtthat X2 4.t~ a t 0=
(s) Ins V:all (1 '2r3 4)Such that '2xt1 + 2 x3 + x =4,
2. Decide which of the followsing are' lin ar arietie ot>f"(P.
(a) Thle set.X of polynom)rials Ax)such that1A() =1.A'(Q)t=t0,
(b) 'The set Y of polynomials 13(x) such that 1(1) = 1$(2)! = 3.
(c) The set Z of polynlom-tials with ntonzero conlstantt term.
3. Decide which of the follow ing; are linear vantries of the spa C of real funtctionts
