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Linear Algebra 2270 Midterm 2 Group A 07/09/2015 Solve 4 1 Find out of 6 problems: krne1 ;incl till range of till following lin ar transformation T : ti1( r’f/ .r x 2 .t’ —* 1 + 2.r — •r ) = J I 2. Let U iw a suhspac(’ of F cl(’1un’d as lullows: IV= { [‘] c’ :.r . 1 .r2 Ill E +.r2 Enrich a set To a hasis of )j/. . ) Is there a Iin(ar t r nsfrinat ion T : transiorinat ion exists, is it uinquv.’: 1’ ([]) 1. Prove the tlieoreiii that if T 5. Let I h1 = V 1 )( two sllhspa((’5 (If using two V(’(t (irs ?Ii e 1 TV /2 [f]. —* 2 sueli that the following is satisfied? F T ([]) aboVe a veutor spa(e represeiltat 11111 Is lflHqlle V —+ .‘ ([]) V Assiiiiie that all !1I + = [1] V((t( rs .r 112 ii amid I )lIly ii T1 fl Tl’2 II’. Write (lVtHhitIuIis ol (a) T is linear is [] 2 as E TV 6. Let 1. T4 is’ two v((’tor spaees amel T (h) T = If sueh a TV is 1—1. onto and imear, its iliverse is also lill(ar. = Prove that the 1 1-i (u) I is onto 1 = O} may he written rn x \J o o •1 o — — I I o o I 1- — 0 I - 4, r’. t I’ 0 ‘I —4 c---_____) — 7Th x. 1- r---- () 3 c) :3 - O r— c. — I) c J2 4 c—. I) c Q L ci ‘1 4 ‘•x (I U U 0 I L 1 — .zzz (—I —s •— —j’ - L ci - •E - 4 tJ — 0 I S - N — - — 0 — c—;----------- -—-----___i_ - 0 — ‘ 0 Ci - CO — — c 1- ci - I-, + Il r. ‘-%‘ r , j -, - H r—, I, - j I I r - (N - 4- c$__, -‘ ‘ 3 1- - N N cc cc -4- J cc II cc 4 l S ‘ 2 C 0 1- I’ S - N - n -- 0 0 I — 4” CC 4-, N t’__ ‘-C ‘ .-c ‘4, ..z N — — p cc cc Th cc -4- c-c — c_c — t%4 Ij -4’ cç N 4’- cc 4— N cr cc 1 V “3 —.4 N (1 c -S..’ cc 1 Lc H Lin’ar Algcl ua 2270 i\Ii(11(91I1 2 G1ouJ) 13 07/09/2015 Solvc 1 out of 6 prob1enu-: 1. Let I V be a siihspace of fl defined as follows: T1 = {[] e 2 , 1 :.r , .r 33 ..i E +,2 — 213 o} —i Enrich a set To a basis of 117. 2. Find the kernel and the range of the fbllowiug linear transformation T : 2 / f L 3 J ([]) 1. Prove the theorem that [ also lniv;ir. : I/ = .1 ([]) is unique (i Let V II Ia’ two vector spaces. Write (h) V.IV Ti T ([h]) ii Assmne that all vectors Y] + !12 and oniy if Wi fl l’V definitions of If such a = V. L : 11, —k V are linear, then the = (a) (lHfl(/) .13 []. Let 1/Vj. I’V he two suhspaees of a vector space V. lisin two vectors ?/l lI i/i E W as Prove that the above represelitatiomi — D such that the following is satisfied? T = + 2x .11 + Zr2 / 3. Is there a linear trrnsformat ion T : transformation exists. is it uinque?: — {}. coniposit .t V hoi ? a T 1 is may Ia’ written _‘\j S.—’ Si “.4 2 “ cS . e:L L .0 3 --Th I’ C’ 5-. - L —., - :— J C, - cJ ‘-I 2 1 -1 -4- V .4- z c) — C.’ C a