Question

Determine whether each of the following is a subspace of the relevant R". (a) V1 = {(x, y, z) | x, y ER, Z E Z} (b) V2 = {(2,4,4) + s(1, 2, 2) + t(4,5,7) | ste R} (c) V3 = {(a, b, c, d) | a, b, c, d e R, ab = 0}

Answer 1

ay Here let ={ < <m, y, z) :x, JER , ZE ZE}. x = 42, 8, 2) and ß = {2, 2, ? 2) Vi ne, &, G, ER 2,2, EZ, belong to Then Nowy Lt ß- <h, 2) + (22, I2 , 2₂) olma, y ty, 27 +22) the Elk YtZ EIR and get , & Z. xß V, & tp t v SEVI KGIR and x = {0,3,3) EV. Now, ka - k<, 2, 3) = {X4, Kgg., K2). K EIR KHEIR But if we Z, = 2 & 2 K= 2 EIR K Z 2 & 2 Jake and ka then

Therefore, egration (2) and (3) * KEIR, LEV, A KREN, Hence, Vi is not ro Subspace of BB IR² b) were = {(2,443*341,2723+72458) Here, ={<2,5,454 8<1,2,2)444435,7) ; ster), (2, 4,4) & 341,2,2) + + (4,5, 7) =<0,0,0) <2+3+4+, 4+25+5+, 4+28+74) =<0,999 which gives, 2 est ht=0 - - {U 4+25+5 t = o-- (2) 4+25+7f=0 -- (3) we get (4+25 +5+) -(4+ 25 +76)=0 5 7 If 20. let Then, from ar

or, - at o ar put the value of tao im (2) weget, 472s ao Si= -2 .: <o,o, os = 22,4,4) - 241,2,2) 07o. (9,5,7) This shocus that <0,oost V2. let Then and L, ß E V2 x = x2,4,4) &S, <1,2,27 + t, 24,5,7) P = 42,4,45 +5₂ <1, 2, 2) +€ ₂4.4,5,7) for some S ,S₂, t , tg EIRI idth=<2,4,47 + 5, (1, 2, 2) + t, <4,5,7] of <2,4,4) + S221, 2, 2) + tz 24,5,7) 242,4,4) + 6 <2;4,4) + (st) {1,2,2% + Cdittz) < 4,5,7) 2<2, 4,47 + (Sth) < 1, 2,2) + (1,1b) 24,5,7) & {2,4, 4) + (5, +sc) <1,42) + (tits) (5,5,7)

This shows that, a, bil but X, B E V2 xaß & V. is not a Supspace of cn3 Hence, as 3 = Z (a, b, ed {a,b,c,d): a, b, c, d E IR, ab of Obviously <0,0,0,0) eV &&- (0., 6.,, on) and ß = [ar, bz, Cr, de) belong to V3. az by so now, let dobro Then a, b, zo Xtß a, de) ka, b, a, et (az, bz, ez, dz) La, da bi + b2, Gtes, detde) (a, ota) (6, +62) = ab, & aby & asb, ot azbe - of aabetarbeto aby eas 6 tor

Therefore, but X, BE V₂ x + ß & No Subspace V3 is not Hence / of 1RY,