Question

(6 pts) 3. Prove whether or not the set of ordered pairs of the form (w ) is a subspace of R'. (6 pts) 4. Prove whether or not the set of polynomials of the form a+Ox+ax is a subspace of P. 5. Let S = {(2,2,1,6),(1,3,5,1),(1,7,14,-3)} . (6 pts) Is S linearly independent or linearly dependent? (3 pts) b. Find (4 pts) 6. Give three examples of vectors in the span of {0.2.1),(3.1.0);

Answer 1

//. solution:- NOW, Let, T be the subset of 1R² defined by T= 3 (USU): UER} het, X, BET Then, 7 ", VER such that 1 xa(u,u?) & Ba(0, 02) Now, x+B a (U, U²) + (0,02) = (utv, u tip2 m (utuje may not be equal to u²+U2 for example, net, x= (1,1) & B=(2,4) Then, both dißET Now, X+B= (1+2, 1+4) 2 (3,5) Then, 3²2975 (275) T Thung X, BET but X+B&T. Therefore, t is not a subspace at IR?. The set of ordered form (1,02) is not a Bairs of the subsface of R?

hetg 3 T be T be the subset of Pq & de fired bil, T= { a+03+022; AC AM] Now , Let, coat het, daatontana & Babtoon&bar Then, daßET heby eGR Then, 1 ateß x teß = atoon tan2te.(6+01+672) = (a +eb) +0.04 + 10 +eb)22 Now, a, b, c ERR a, b, et г) а+eb ER Thus, Kreß = (areb) to .x + (a+b)x² ET " X, BETT & CER => d+eBET. Therefore, Tis a subspace of P2. The set of Balynomial of the form / a +0% +ax2 is a subsface of P2

salution :- () را ر 3 , 571 را) ) ,( 1 , 1 , 14 3- , ) ده ولها | ( وار 2 , 2 ,1) = 82 3 , 5 (ا, 83 = (1,1,14, -3) Letus consider the relation, CIR 3 ررر> ; 20 ه ح +20 +18 صدا ) 2 + (کرار 2 ,2 ) ۱ , 3 , 5 + (ا, | 0 = (3-ر14ر1 را) ۱۰۴ رو 1+52+ رم ر763+32 +20 ,23 +1 + ع2 ) زه ره ره ره) = (3 3- 1) + (ع) Thw 2 , +2+3 20 – 2 ع ) . + 5 - 0 = وع7+32+ 0 23 - 2 c + + 1 =0 - 3 5 0mh From , * - 2(+32+ 3 - 2 - 2 - 3 = 6 => 2 ) + 3 =5 (0 . ے و3- = . في - x3 2 , ۴۲۵۳ ا (+3+ 3 - 2 - 3 - 7 = 0 = H - 2 =0 عم =e. ج-

. we see that, ei aeg & C22-3e3 satisties W, 6. It, we take 13-1 Thin, e, al & e2=-3 Then from o 21-322 +43 = 0 Therefore, Qiyal, az are linearby defiendent. s is linearly defendent fone, solution:- since, s contains B rectors but sis linearly defendent Therefore, dim ( span (s) ) <2. & we see that, any two vectors of s is not multiple of another. Thus, dim i span(s)) = 2 .. dim (span (s)) =2 /

solution : hela و رورارو ) , ( 1 , 271) =3 (۱, 1 , 2) ده رية امرار 3 ) » . ۱ ر ره ز 2 + 18ع = (3) spen * ا= 2 & ادم رعاع و علما ,If ا , TRY رور ر2 ) ۱۰ + رار 1 , 2) .) : 8 2 2+ ,2 | (6+را+ 2 ر2+) 2 (هر 2 ) - If, we take e, al & eq=-1 + c را زرار ) - (ار,۱) و 2 رها ,ا-2 ,3 -۱) زارار- )= e, 22,c2=1 take It, we 2+ ,8 ,2 , 1 زہرا,3) + راہ,۱) .2 , (+1 , (+ہا , 3+1 ) 2 هر5 ,5) - in the span of three veeters Examples of رارار 2-) زه رد ,ا) در (ه دار 3),(2 , 1رن هر 5ر5) 5