Midterm 1 - 2014-15 Spring.pdf - METU NCC LINEAR ALGEBRA...

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Unformatted text preview: METU - NCC LINEAR ALGEBRA MIDTERM EXAM 1 Code I MAT 260 Last Name: Acad.YearI 2014-2015 Name I Semester l SPRING Student # : Date i 22.03.2015 Signature : Time ‘ 09’4" 4 QUESTIONS ON 4 PAGES Duration: 90 mm ' TOTAL 100 POINTS IIII—J-U 1. (15pm) Let U = {(3 a:,,y,w z) : ~22: + 33,— w + z = D} C R4. Find a. basis for U Justify your answer, that 15, show that the set you find IS linearly independent and it spans the subspace U. ,m : 26,33”, 'an’sfim); K231“) (—422 Team E : i (Maia/2.), (ergo “5);(@;0:\,\) C TMR 1:5? “QC ): M be. Qw‘fie. (mus/w i1fi~3~51wj2 xfl‘mfllofl.) "h3(b1\90 “Bl-i ago/0,1”) Nerdy E>~A??¢3R VKQI‘WOEOIE) ”R‘FLQ O.\)Oam‘5) ‘i‘ r$<ore?§3‘>': (fife/gag) imc‘k. & E LS ox \ootsm (341M, 2.(15+15pts) Determine if the following sets are linearly independent. Justify your answers. (a) E = {x +1, 32 — 1, 1:3 + $2.+$} Q 733(k). $3433}??? 1“ 01-170 4139; §-\ 1K1) AchgJCX-a'xl-rfiq’) 5: C) “Hear-10.2 E 13 \‘Mn r1\’\&\' (b)E={(1,1,2,——1), (1,0,1,0), (2,—1,1,2)}c;.1122.4. “new Y'V-W‘a 424}; :- Q m “V3.3: C3 Zn «r2? W3) TC) (*5: (imp/m :0 *M *Z‘F‘scg ("2:11) 3.{25pts) Let 'E = {23¢I + Xb — Xe, xa — 290, + x6} ; Rm(S), where .S‘ 2 {a,b, c}. Find the subSpace Span(E) spanned by E. ‘) <31, ()4) +/‘<L"“«2, 1) 2: 0, (1,2) W . 3’11‘7‘5’\ _, x +35 +5220 50, SWLE) :JL in) +33%) +539(c):oj . ‘5' 4.(30=10+15+5pts) Consider subspaces U = {11—m2+2x3—m5 = 0} and V = {x1 +xg—2x4 = 0} of the vector space R5. (a) Find the dimensions dim(U) and dim(V) of these subSpaoes. Justify your answer. (Bu :5; i, 4) 0,0,0>,<~3, 0,1,0») (a, o, 0, Log <41, 0201621)) Am ('14) = LI @ ix: L054, 0,0,6», <3, 0,- 0, be») (0,62, {£01 0>j < 0', 0, 0, 6.2, DJ WV): q s ‘ ' (b) Find the dimensions dim(U n V) and dim(U7+ V). Justify your answer. um i “hhmfiefiso M" ””3 1311945 3’0 i . x‘ Hz, «gm, :0 (31 yzg “31*; flax?» x5 2 o 13, my; 1cm; in, :> (u (w 3.: Wyn-T!) (c) Does the equality dim(U + V) + dim(U {‘1 V) = dim(U) + dim(V) hold? 3+5 :q‘Jr-HE; ...
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