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08win-m1sols - Math 51 Winter 2008 — Midterm Exam I...

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Unformatted text preview: Math 51- Winter 2008 — Midterm Exam I Please circle the name of your TA: Zachary Cohn José Perea Nikola Penev Man Chun Li Daniel Mathews Theodora Bourni Anssi Lahtinen Isidora Milin Circle the time your TTh section meets: 10:00 11:00 1:15 2:15 Your name (print): , UT Student ID: Sign to indicate that you accept the honor code: ‘mtmsz mm: A a 4 Instructions: Circle your TA’S name and the time that you attend the TTh section. Read each question carefully, and show all your work. You have 90 minutes to do all the problems. During the test, you may NOT use any notes, books, or calculators. Question Maximum Score Formulas you may use: 351 3/1 552% — 173% 3333/1 — 131313 $3 93 $1y2 — 53291 H to X {Q N | Problem 1. (12 points total) (a) Write the equation of the line passing through the point (2,1) and with normal vector n:(3§. J22 :1— fa) r: O W (12:11“) O ACHvL) + 11A r 0 ’ x i + x l a w l (b) What is the parametric equation of this line? K 3 :2. i ‘lfl: “E 'm“““"“- l Umfikgrg XL : "E: p w Problem 2. (12 points) Given two vectors u and 11 such that = ShOW that u + ’U and u — v are orthogonal (perpendicular). (Hint: Use dot product) (jumilgak’Wwi) : “LLth "mg/v 1+ {ELI/Ki wv’av’ ,, 4.. and?” w H --*-~ 0 swllwll W“ 30 Merv cw,ch «v v' cwc alliwymkk Problem 3. (12 points) Consider system of equations x+y=2 x+ay=b. Where a and b are constants, and :1: and y are the unknowns. Determine all the values of a and b for which the system above has: (a) no solution CL, ‘3' 1 L) 34’: 2... (b) a unique solution OK if l (o) exactly two solution "M QM (d) more than two solutions (2w 3- l , l3 5 L Explain your answer below! a 4 1 ‘ 1 l l \ > ._._..> < l l _ ‘\ CL lo 0 67m 11> -2_ Km “W “QM fl Uni 3o tan-t va #0 K Cu (omkxlzkkmi KC: Mmm‘t ,3 Ci M «M C W MEMM, (ACMMQ‘ ‘1? xiii. 9’ $2 Sirens KM we a. :«i , \e 2 2 053C 33%“ l as )4 \ h a) a . | t A h Wfiwfiegflux a mi,2,_ {w Ks, @Lia'mi 0:; WW Aim e.) N; ‘\‘ ’2 381% ‘9 s 1 \v vs) WV, we a lbw m (fwdxk w a +1 ‘ 3 [Wk 3 A NIKKW k” d‘UM/M (j 1:“ ’e ’4 Q )0 *3 Problem 4. (15 points total) Let {u,v, w} be three linearly independent vectors. (a) (5 points) Show that {u + v, u — v} are linearly independent. assume Xiflu+rv3 + “li‘v V) " O GM“- 9\5‘* *\,3(7,) so (xi-erzyw 'l- QL‘IXL3V “0 uJVKfl-x (web/J “no K‘ +12 :O >( 'x_ 1030 6, will 30 x“ '3 O ) XL Mlv , Mrv M QATM Chub (5 points) Are {u, u + v, u —— v} linearly independent? Please explain your answer. N0 3 Kt“ 1? xm,mr/\fi& “2% 4"“1 lM*v) + |(u-v) :O (c) (5 points) Is w in the span of {u + w, v + w}? Please explain your answer. \\\»0 w}? Mfiumflk} W (vi/h '3 can W ":gX‘kaw + X1 RV 4 W) 032.7% J _ i ,7 : - w «Kart» X‘NL 4 )£2 v +_‘()‘L\+xz~\)w Q “Ag/J (.ci/ Mg 83% 5 ‘W W (‘aMMSX—E‘ (kw chgw Qioe‘ih 5e X10112: QDx‘-+)szl':c> \ it Problem 5. (15 points total) Consider the matrix A = ( 1 1 —1 1 > (a) find a basis for N(A) V \ 4 l @(1\~:l“)W>L®\ 0 I) \\\\) 0020 00®o XZDXLI % 3 2*»); X1: x5 T, y“ A X1~X‘I *1 ’l >( x i X7- : 2» 3 +X‘l 3 O 2 O o Xll X“ Q -l 'e-l S - Q VI J ‘ JL 2 0 &mW u [OMQQ N G D ’1 (b) find a basis for C(A) law EA SfiQMJ %\@w \G’QM’V‘KM tag (wwnffik (LLX l L) Siam 5.. lav-3Lan (3A) 2 1 . 0 (c) glven that A 0 —_ < 2 1 ), find all solutions to the equation A33 2 < 3 > Problem 6. (12 points total) Which of the following are linear subspaces of R2? Please explain your answer. (a) the set V = E R2 | :131 S 0} u maxim 3&36LQCL PA New X 11(2) ‘5 “N 13“} en t L‘) 9 mi “v V O (b) the set V = {($1,332) 6 R2 I 5131: 0} \fxesfilxs Ck 3%» VQUL, an. WW Cl" u; e4 we ‘3qu A 2 km) ‘ ii (eengl who Che“ K524 ) Problem 7. (12 pts total) Consider the following linear subspace of R3: V = {($1,1J27IL'3) '331’“ $2 + 1133 = (a) find a basis for V “(A . ' ml ‘ 30 U“: l} UL 1L0 ) a O ‘ l (b) give an example of a matrix A such that N (A) = V. R am ,--\ , l) (0) give an example of a matrix A such that C’ (A) = V. Problem 8. (10 points) Circle T or F to mark each of the following true or false. Explanations are not required for this problem. (a) the dot product of two vectors in R3 is a vector in R3 T (1?) H0 ) in u ‘3 (disk; (f) any three vectors in R3 span R3. T ® Ho 3 w Ufikm €®u§cx \x X?u&, (b) a system of 3 linear equations with 5 unknowns cannot have a unique solution. @F L310) we “My «'1 v ItQpJWKCx‘GQ‘LQ ‘53 (Gammék mu; W" (c) a system of 5 linear equations with 3 unknowns cannot have more then one solution. T® big“) Ski-“Q‘S‘L‘ZA > QM g3 ‘22 (Gagei LN QC} M. coma, (d) for all matrices A, the column space of A equals the column space of the rref (A) T “mo 5" 6%; Qavol’affivu 500,) r (e) for all matrices A, th'e null space of A equals the null spacelof the ' F J’Wxs Eek :G‘ «2;, {WEM‘WVVQ {vwk iflfltm “~53 7‘ (g)ifAisa4><2matrixl endimN(A 32 @F i} shim MUM ‘3" “as, Q. 2. “$25th 7‘? was (h) if A is a 2 x 4 matrix then dim N(A) Z 2 d, ’ V a Q 4 a? "so “i {i , . a‘ i: {M l Q; s GUa year” 4 U 4”” ~ ' l a (i) there are 3 X 6 matrices» with dim N(A) = 3 and dim C(A : 3. ) . "We; (ti§:§é§>m“mmwl’ » 0-04 5‘33 (:39 Q (j) there are 6 x 3 matrices with d1mN(A) = 3 and dim C(A) = 3. T 6) Nojm w umfifla I) 50 Mg was“, \m “)9me (:th "5 six um ...
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